Combinations Formula with Examples Last Updated : 13 Dec, 2024 Comments Improve Suggest changes Like Article Like Report Combinations are way of selecting items from a collection of items. Different groups that can be formed by choosing r things from a given set of n different things, ignoring their order of arrangement, are called combinations of n things taken r at a time.The number of all such combinations is calculated by the combinations formula: nCr or C(n, r).Example: All the combinations of four different objects a, b, c, d taken two at a time are ab, ac, ad, bc, bd, cd. Here we have not included ba, ca, da, cb, db, and dc as the order does not alter the combination. Thus there are 6 combinations of 4 different objects taken 2 at a time i.e., 4C2 = 6Similarly, all the combinations of four different objects a, b, c and d taken three at a time are abc, bcd, cda, dab.Thus there are four combinations of 4 different objects taken 3 at a time i.e., 4C3 = 4Corresponding to each of these combinations, we have 3! permutations, as three objects in each combination can be arranged among themselves in 3! ways. Hence, the number of permutations = 4C3 × 3!4P3 = 4C3 × 3!4!/(4-3)! 3! = 4C3Thus we can conclude that the total number of permutations of n different things taken r at a time i.e., nPr is equal to nCr × r!Hence,nPr = nCr × r! , 0 ≤ r ≤ nThis implies,nCr = n! ⁄ r! (n-r)!Formula for CombinationsThe combinations formula provides a way to calculate the number of combinations of n different things taken r at a time is given bynCr = n! ⁄ r! (n-r)! ,0 < r ≤nwhere,n is the size of the set from which elements are permutedr is the size of each permutation! is factorial operatorRelation between Combinations Formula and Permutations FormulaThe main difference between combination and permutation is only that in permutation we also consider the order of selecting the things but in combination order of selection does not matter. And therefore, permutations are always greater than the combination. Theorem: nPr = nCr × r! ( Permutations formula = Combinations formula × r! )Proof:Consider,RHS = nCr × r! ⇒ RHS = [ n!/ r!(n-r)!]r!⇒ RHS = n!/(n-r)! = nPrHence, the theorem is verified.Read More about Difference between Permutation and Combination.Key Points to Remember for Combination FormulaWe have nCr = n!/r!(n-r)! In particular, if r = n, then nCn = n! /n! = 1nC0 = n! /0! (n-0)! = n!/0!n! = 1/0! = 1. Thus the formula nCr = n!/r!(n-r)! is applicable for r = 0 also.Hence, nCr = n!/r!(n-r)! , 0 ≤ r ≤ nnCn-r = n!/ (n-r)![n-(n-r)]! = n!/ (n-r)! r! = nCr.Hence, nCr = nCn-r i.e., selecting r objects is same as rejecting (n-r) objectsRead More,Permutation and CombinationAptitude Questions for Permutation and Combination Tricks to Solve Permutation & Combination QuestionsSample QuestionsQuestion 1: From a class of 30 students, 4 are to be chosen for the competition. In how many ways can they be chosen?Solution:Total students = n = 30Number of students to be chosen = r = 4Hence, Total number of ways 4 students out of 30 can be chosen is 30C4 30C4 = 30! / (4!(30-4)!)⇒ 30C4 = 30! / (4!26!)⇒ 30C4 = 27,405 waysQuestion 2: Nitin has 5 friends. In how many ways can he invite one or more of them to his party.Solution:Nitin may invite (i) one of them (ii) two of them (iii) three of them (iv) four of them (v) all of them with combinations formula we can calculate that this can be done in 5C1, 5C2, 5C3, 5C4, 5C5 waysTherefore, The total number of ways = 5C1 + 5C2 + 5C3 + 5C4 + 5C5= 5!/ (1! 4!) + 5!/ (2! 3!) + 5!/ (4! 1!) + 5!/ (5! 0!)= 5 + 10 + 10 + 5 +1 = 31 waysQuestion 3: Find the number of diagonals that can be drawn by joining the angular points of an octagon.Solution:A diagonal is made by joining any two angular points. There are 8 vertices or angular points in an octagonTherefore, by using combinations formula the Number of straight lines formed = 8C2 ⇒ 8C2 = 8!/ (2! 6!)⇒ 8C2 = 8 ×7 / (2 × 1)⇒ 8C2 = 56/ 2 = 28Which also includes the 8 sides of the octagon.Therefore, Number of diagonal = 28 - sides of octagon = 28 - 8 = 20 diagonalsQuestion 4: Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls, and 7 blue balls, if each selection consists of 3 balls of each color.Solution:Number of Red balls = 6Number of white balls = 5Number of Blue balls = 7Thus, Total number of balls to be selected = 9Hence, the required number of ways of selecting 9 balls from 6 red, 5 white, 7 blue balls consisting of 3 balls of each color can be calculated by using combinations formula = 6C3 × 5C3 × 7C3= 6!/ (3! 3!) × 5!/ (3! 2!) × 7!/ (3! 4!)= 20 × 10 × 35 = 7000 ways Comment More infoAdvertise with us Next Article Fourier Series Formula A anurekha1500 Follow Improve Article Tags : Mathematics School Learning Maths-Formulas Similar Reads Basic Math Formulas Mathematics is divided into various branches according to the way of calculation involved and the topics covered by them. All the branches have various formulas that are used for solving various mathematical problems. The branches include geometry, algebra, arithmetic, percentage, exponential, etc.T 12 min read Number SystemWhat is the Division Formula?Division is one of the top four important arithmetic operations (i.e., Addition, subtraction, multiplication, division). Division operation is used to split the number into equal parts. The symbolic representation for division is '÷' and '/'. a divided by b can be represented as a÷b or a/b. The form 4 min read LCM Formula | Examples & Practice QuestionsLCM stands for Least Common Multiple. LCM of two numbers say a and b is defined as the smallest positive integer divisible by both the numbers a and b. Hence, the LCM is the smallest common multiple of two or more numbers. It is also called the lowest common multiple, or smallest common multiple.In 6 min read Distributive Property | Definition and ExamplesDistributive Property in Maths refers to the distribution of the number over the other operations. It is usually called the Distributive Law of Multiplication over Addition and Subtraction as the number which is to be multiplied is distributed over addition or subtraction of the numbers.In this arti 8 min read ExponentsExponents are a way to show that a number (base) is multiplied by itself many times. It's written as a small number (called the exponent) to the top right of the base number.Think of exponents as a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing t 9 min read Consecutive IntegersConsecutive Integers are the integers that follow each other, i.e. while continuously writing integers they come next to each other. they have a difference of one(1). For example, ...-3, -2, -1, 0, 1, 2, 3,... this is a sequence of consecutive integers. Apart from that natural numbers are also calle 9 min read Scientific Notation FormulaScientific notation is a way of expressing very large or very small numbers in a compact form. Instead of writing out all the digits, we write a number as a product of a number between 1 and 10 and a power of 10.In this article, we will discuss what scientific notation is, its formula, rules, uses, 5 min read Binary FormulaBinary formulas are formulas that are used to convert binary numbers to other number systems. A binary number system is a system of numbers that has a base of 2 and uses only two digits, "0 and 1". It is one of the four types of number systems and is most commonly employed by computer languages like 9 min read Convert Binary fraction to DecimalGiven an string of binary number n. Convert binary fractional n into it's decimal equivalent. Examples: Input: n = 110.101 Output: 6.625 Input: n = 101.1101 Output: 5.8125We strongly recommend that you click here and practice it, before moving on to the solution. Following are the steps of convertin 6 min read Fibonacci Sequence FormulaFibonacci Sequence Formula: Fibonacci sequence, the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, …, each of which, after the second, is the sum of the two previous numbers; that is, the nth Fibonacci number Fn = Fn − 1 + Fn − 2.In the Fibonacci sequence, each number in the series is calculated by a 7 min read ArithmeticDirect Variation: Definition, Formula and ExamplesDirect Variation is the relationship between two variables in which one is a constant multiple of the other and one changes according to the other. For example, if 'a' is directly varied to 'b' then 'a' and 'b' change accordingly. In this article, we will learn about Direct Variation definition, Dir 7 min read What is Celsius Formula?Answer: The formula for converting temperature from Celsius to Fahrenheit is (9/5 × °C) +32.The Celsius formula is used to convert temperatures between Celsius and other temperature scales like Fahrenheit and KelvinTemperature significantly can be defined as the measure of how atoms are moving and c 6 min read Fahrenheit to Celsius (°F to °C) | Formula , Conversion and ExamplesThe Fahrenheit to Celsius formula defines the method for conversion of the temperature from Fahrenheit Scale to the Celsius Scale. Both scales measure the degree of hotness or coldness of any object or body. In our daily lives, the Celsius scale is commonly used for reporting temperatures in weather 8 min read Revenue FormulaRevenue is the total revenue gained via the sale of goods and services by the enterprise. Thus, revenue is the most significant aspect to define the improvement of any company. The revenue formula is utilized to find the revenue. We estimate the revenue by seeing the outcome of the number of sales a 5 min read Selling Price FormulaSelling Price Formula: Selling price is defined as the amount a customer pays for a product or service. Every day we purchase one thing or the other, and every product has a certain price. So, the price that we pay to the seller to buy the product is called the "selling price," whereas the amount at 9 min read Simple InterestSimple Interest (SI) is a method of calculating the interest charged or earned on a principal amount over a fixed period. It is calculated based solely on the principal amount, which remains unchanged throughout the calculation.Simple Interest is widely used across industries such as banking, financ 9 min read Compound InterestCompound Interest is the interest that is calculated against a loan or deposit amount in which interest is calculated for the principal as well as the previous interest earned. Compound interest is used in the banking and finance sectors and is also useful in other sectors. A few of its uses are:Gro 9 min read Monthly Compound Interest FormulaThe monthly compound interest formula is used to compute compound interest every month. Compound interest is sometimes known as interest on interest. The first period of compound interest resembles the first period of simple interest, but the second period is distinct. Interest is computed on the in 8 min read Daily Compound Interest Formula with ExamplesCompound interest is interest earned on both the principal and interest over a specific period. The interest that accumulates on a principal over time is equally accounted for as the principal. Furthermore, the following period's interest calculation is based on the cumulative principal value. It is 6 min read Double Time FormulaOne needs to know the Double Time Formula to make an accurate estimation of time in project management, resource allocation, and assuring that things are running according to plans. One implements this formula to find out exactly how much time it will take for a particular job if its resources are d 6 min read Basic GeometryPerpendicular LinesPerpendicular Lines in Mathematics are pairs of lines that always intersect each other at right angles, i.e. perpendicular lines are always intersect at 90°. The perpendicular lines are readily seen by us, the corners of the walls, the corners of the desk, and others represent the perpendicular line 13 min read Right AngleA Right Angle is an angle whose measure in degrees is 90 degrees and its measure in radians is Ï€/2 radian. It is one of the most basic angles in geometry and has various applications. It is used to define various shapes, structures, and figures in 2-D and 3-D space. Two lines that are perpendicular 10 min read What is Parallel Lines Formula?The combination of two or more lines that are stretched to infinity and never intersect each other is called parallel lines or coplanar lines. The parallel lines are denoted by a special symbol, given by ||. Properties: Two lines are said to be parallel if they maintain a constant perpendicular dist 5 min read Angles FormulaAn angle is the space formed between two intersecting lines or rays. These two lines are called the arms of the angle, and the point where they meet is known as the vertex. Angles are fundamental elements in geometry and are measured in specific units—most commonly in degrees (°) or radians (rad).An 8 min read Degrees to Radian ConverterConverting degrees to radians is a fundamental skill in mathematics, particularly in trigonometry, calculus, and various fields like physics and engineering. This Degrees to Radian Converter allows you to easily switch between degrees and radians and vice versa. Degrees to Radian Converter This calc 4 min read Area of 2D ShapesArea is the amount of space inside a shape. It tells you how large the shape is and is measured in square units like square centimeters (cm²) or square meters (m²). The area of a 2D shape is the space it covers. Each shape has its own formula for calculating area. For example, you can count how many 7 min read Area of QuadrilateralA quadrilateral is a polygon with four sides, four angles, and four vertices. It is a two-dimensional figure, formed by joining four points (vertices) with straight lines (sides). The side of a quadrilateral may or may not be equal. Various types of quadrilaterals can be defined based on the propert 9 min read Area of SquareThe area of a Square is defined as the space enclosed by the boundary of the square. Measurement of the area is done in square units. The unit for measurement of the area is m2.Let's understand the Area of square with the following illustration:To calculate square's area we need to know the length o 7 min read What is the Diameter Formula?Answer: The formula for the diameter is D = 2 × R, Where R is the radius of the circleDiameter is defined as the line that divides the circle into 2 halves i.e., each part is known as a semicircle. The center of the circle act as the midpoint of diameter. This means that it divides the diameter into 3 min read Arc Length FormulaArc length is the distance along the curved path of a circle or any part of its circumference. We define arc length as measuring the length of a slice of pizza crust. Arc length is calculated using the simple formula: Arc Length= r × θ where 'r' is the radius of the circle and 'θ' is the angle in ra 7 min read Central Angle of Circle Formula with Solved ExamplesCentral angle of Circle is the angle formed by two radii that meet at the center of the circle. This angle helps us understand the relationships between different parts of the circle. Knowing the central angle is useful in many real-life situations, such as designing circular objects and calculating 5 min read Asymptote FormulaIn geometry, an asymptote is a straight line that approaches a curve on the graph and tends to meet the curve at infinity. An asymptote is a line that a graph of a function approaches but never touches or crosses as it extends towards infinity or a specific point. Asymptotes help to describe the beh 8 min read Axis of Symmetry of a ParabolaA parabola is a U-shaped curve that is the graph of a quadratic function. In mathematical terms, a parabola is the set of all points in a plane that are equidistant from a fixed point called the "focus" and a fixed line called the "directrix."For example: Consider the quadratic function y = x2 − 4x 7 min read Centroid of a Trapezoid FormulaA trapezoid is a type of quadrilateral with two parallel sides. A quadrilateral is a type of polygon with four sides. The sum of the four internal angles of a quadrilateral is 360°. Centroid refers to the center point of any figure. It is also known as the geometric center.The centroid of a trapezoi 7 min read Area of a Circle: Formula, Derivation, ExamplesThe area of a Circle is the measure of the two-dimensional space enclosed within its boundaries. It is mostly calculated by the size of the circle's radius which is the distance from the center of the circle to any point on its edge. The area of a circle is proportional to the radius of the circle. 10 min read Parallelogram FormulasGeometry is a field of study which it is about shapes and their properties. The shapes are studied on the basis of their dimensions, composition, angles, and other properties. The field deals with the calculation of area, volume, perimeter, and other parameters of a particular shape. What is a paral 4 min read Perimeter Formulas for Geometric ShapesPerimeter formulas are used to calculate the total length around any geometric shape. Geometry is all around us, from everyday objects to buildings, and understanding the perimeter is essential in many practical applications.In mathematics, the study of shapes and their dimensions is called mensurat 6 min read Perimeter of TriangleThe perimeter of a triangle is the total length of its three sides. A triangle is a polygon with three sides, three vertices, and three angles. It is the simplest closed polygon in geometry, as it is the first possible closed figure. Any polygon can be divided into triangles. For instance, a quadril 5 min read Equilateral TriangleAn equilateral triangle, also known as a triangle with equal sides, is a fundamental shape in geometry. Each of its sides is of equal length, and its interior angles are all 60 degrees, making it a 60-degree triangle. It is also a perfectly symmetrical shape. In the figure given below, ∆ABC is an eq 9 min read Scalene Triangle: Definition, Properties, Formula, ExamplesScalene Triangle is a type of triangle where all three sides are different lengths, and all three angles have different measures, a scalene triangle is unique in its irregularity and it does not have any symmetry. Classification of TrianglesWe can classify the triangles into various categories by co 6 min read Right Angled Triangle | Properties and FormulaRight Angle Triangle is a type of triangle that has one angle measuring exactly 90 degrees or right angle (90°). It is also known as the right triangle. In a right triangle, the two shorter sides called the perpendicular and the base, meet at the right angle (90°), while the longest side, opposite t 6 min read Perimeter of RectangleA rectangle is a two-dimensional plane quadrilateral, with opposite sides equal and all four angles equal. The perimeter of a rectangle can be defined as the sum of the length of all four sides in a rectangle.Perimeter of rectangle is the total length of the boundary or the sum of all its sides. In 7 min read What is the Formula for Perimeter of a Square?Answer: The perimeter of a Square is equal to the sum of its all sides i.e. 4a, where a is the length of its side.A square is a rectangle whose adjacent sides are equal in length. This means it is a quadrilateral where all the sides are equal in length. Each of the angles in a square is a right angl 5 min read Circumference FormulaThe Circumference Formula is crucial in geometry, especially when dealing with circles and related shapes. This article dives deep into the concept of the circumference, its formula, and how it is applied to various geometric shapes. You'll also find practice problems to test your understanding and 9 min read Perimeter of a ParallelogramA parallelogram is a type of quadrilateral with four equal sides with opposite sides equal. Its sides do not intersect each other. There are two diagonals of a parallelogram that intersect each other at the center. A diagonal divides the parallelogram into two equal parts or triangles. The following 7 min read Rhombus FormulaUnderstanding the rhombus formula is essential for anyone studying geometry. Mensuration is a branch of geometry that studies or measures the area, perimeter, and volume of two-dimensional or three-dimensional objects and constructions. Mensuration comprises fundamental mathematical formulae and, in 9 min read Perimeter of Rhombus FormulaIn mensuration, the perimeter of a is defined as the sum of lengths of all the sides of the quadrilateral around the border. So perimeter of the rhombus is defined as the sum of all 4 sides of the rhombus.Rhombus is a diamond-shaped quadrilateral whose all sides are equal but each angle inclined bet 5 min read Diagonal FormulaDiagonal Formula is the formula that is used to calculate the length of the diagonal of any polygon. A diagonal is a slant line that joints two alternate sides of any polygon and finding its length is very important for solving various problems of geometry.There are different ways for finding the di 7 min read Diagonal of a Polygon FormulaDiagonals of a polygon are the lines that connect the alternate vertices of the polygon. A polygon of n sides has n(n-3)/2 diagonals. A polygon is a closed figure with n sides (where n is always greater than equal to 3). A polygon is a closed shape with three or more straight sides, and diagonals ar 10 min read Diagonal of a Square FormulaDiagonal of a square is the shortest distance between the non-adjacent vertices of a square. It is a fundamental concept in geometry, often used in various mathematical problems and real-world applications. This diagonal splits the square into two congruent isosceles right triangles, providing a bas 5 min read Diagonal of Parallelogram FormulaDiagonal of Parallelogram Formula: A parallelogram is a quadrilateral with equal pairs of opposite sides and angles. One of its pairs of opposite sides is parallel to the other. The interior angles lying are supplementary, that is, their sum is 180 degrees. The diagonals of a parallelogram bisect ea 7 min read Diagonal of a Cube FormulaDiagonal of a cube is the line segment joining the two non-adjacent vertices of a Cube. The diagonal of a cube formula helps us to calculate the length of diagonals in a cube. There are primarily two diagonals in a cube, namely face diagonals and body diagonals. In this article, we will learn the ty 8 min read Euclid Euler TheoremAccording to Euclid Euler Theorem, a perfect number which is even, can be represented in the form (2^n - 1)*(2^n / 2) )) where n is a prime number and 2^n - 1 is a Mersenne prime number. It is a product of a power of 2 with a Mersenne prime number. This theorem establishes a connection between a Mer 10 min read What is Side Angle Side Formula?SAS Formula, area = 1/2 × a × b × sin c.In geometry, two figures or objects are considered congruent if they have the same shape and size, or if one of them has the same shape and size as the mirror image of the other. More formally, a set of two points is said to be congruent only if one can be tra 11 min read Polygon Formula - Definition, Symbol, ExamplesPolygons are closed two-dimensional shapes made with three or more lines, where each line intersects at vertices. Polygons can have various numbers of sides, such as three (triangles), four (quadrilaterals), and more. In this article, we will learn about the polygon definition, the characteristics o 7 min read MensurationAnnulus Area FormulaThe area of an annulus is the difference between the areas of the two circles of radius R and r.R is the radius of the bigger circle, while r is the radius of the smaller circle. The common area between the two circles is known as the annulus region, and the formula used to find that area is known a 7 min read Volume Formulas for 3D ShapesVolume refers to the amount of space occupied by a three-dimensional object. In geometry, calculating the volume is essential for understanding the capacity of a shape. It is used in various fields like engineering, architecture, and manufacturing to determine the amount of material or space an obje 7 min read Volume of a CubeVolume of a Cube is defined as the total number of cubic units occupied by the cube completely. A cube is a three-dimensional solid figure, having 6 square faces. Volume is nothing but the total space occupied by an object. An object with a larger volume would occupy more space. The volume of the cu 9 min read Volume of a Cylinder| Formula, Definition and ExamplesVolume of a cylinder is a fundamental concept in geometry and plays a crucial role in various real-life applications. It is a measure which signifies the amount of material the cylinder can carry. It is also defined as the space occupied by the Cylinder. The formula for the volume of a cylinder is Ï€ 11 min read Volume of Cone- Formula, Derivation and ExamplesVolume of a cone can be defined as the space occupied by the cone. As we know, a cone is a three-dimensional geometric shape with a circular base and a single apex (vertex).Let's learn about Volume of Cone in detail, including its Formula, Examples, and the Frustum of Cone.Volume of ConeA cone's vol 9 min read Volume of a SphereThe volume of a sphere helps us understand how much space a perfectly round object occupies, from tiny balls to large planets. Using the simple volume of sphere formula, you can easily calculate the space inside any sphere. Whether you're curious about the volume of a solid sphere in math or science 8 min read Surface Area FormulasSurface area refers to the total area that the surface of a three-dimensional object occupies. It is a measure of how much-exposed area the object has. It’s used in various fields like engineering, architecture, and design to determine the material needed for constructing or covering objects.The met 8 min read Surface Area of a ConeA cone is a 3-dimensional geometric figure with a circular base and a pointed top called the apex. The distance between the center of the circular base and the apex is the height of the cone. The surface of the cone curves smoothly from the edge of the base to the apex. It is a common shape found in 8 min read Surface Area of Sphere | Formula, Derivation and Solved ExamplesA sphere is a three-dimensional object with all points on its surface equidistant from its center, giving it a perfectly round shape. The surface area of a sphere is the total area that covers its outer surface.To calculate the surface area of a sphere with radius r, we use the formula:Surface Area 8 min read Surface Area of a Square PyramidPyramid is a three-dimensional geometric structure with a polygonal base and triangular faces equal to the number of sides in the base. The triangular faces or lateral surfaces of a pyramid meet at a single point known as the apex or the vertex. Surface Area of a Square Pyramid In a pyramid, the bas 8 min read Volume of a Pyramid FormulaPyramid is a three-dimensional shape whose base is a polygon, and all its triangular faces join at a common point called the apex. The pyramids of Egypt are real-life examples of pyramids. Volume of a pyramid is the space occupied by that pyramid and is calculated by the formula, V = 1/3×(Area of Ba 9 min read Frustum of ConeFrustum of a cone is a special shape that is formed when we cut the cone with a plane parallel to its base. The cone is a three-dimensional shape having a circular base and a vertex. So the frustum of a cone is a solid volume that is formed by removing a part of the cone with a plane parallel to cir 10 min read Volume of a Square Pyramid FormulaA pyramid is a three-dimensional polyhedron with a polygonal base and three or more triangle-shaped faces that meet above the base. The faces are the triangle sides, while the apex is the point above the base. The base is connected to the peak to form a pyramid. When the pyramid's base is in the sha 8 min read Surface Area of a PrismSurface Area of a Prism: In mathematics, a prism is an essential member of the polyhedron family and is defined as a three-dimensional shape having two identical polygons facing each other that are connected by rectangular or parallelogram faces laterally. The identical polygons can be triangles, sq 13 min read Frustum of a Regular Pyramid FormulaA Pyramid is a Mathematical figure having three or four triangular faces as sides and a flat polygonal base which can be triangular, square or rectangular, etc. The side triangular faces are called Lateral faces. The common meeting point of all the triangular faces is called the apex. For a given py 6 min read AlgebraBasic Math FormulasMathematics is divided into various branches according to the way of calculation involved and the topics covered by them. All the branches have various formulas that are used for solving various mathematical problems. The branches include geometry, algebra, arithmetic, percentage, exponential, etc.T 12 min read Algebra Formulas - List of all Algebra FormulasAlgebra formulas are mathematical expressions that help solve problems involving variables and constants. They often represent relationships between quantities and can be used to simplify calculations. This article provides a comprehensive overview of all algebra formulas taught from Class 9 through 10 min read Polynomial FormulaThe polynomial Formula gives the standard form of polynomial expressions. It specifies the arrangement of algebraic expressions according to their increasing or decreasing power of variables. The General Formula of a Polynomial:f(x) = an​xn + an−1​xn−1 + ⋯ + a1​x + a0​Where,an​, an−1​, …, a1​, a0​ a 5 min read Factorization of PolynomialFactorization in mathematics refers to the process of expressing a number or an algebraic expression as a product of simpler factors. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, and we can express 12 as 12 = 1 × 12, 2 × 6, or 4 × 3.Similarly, factorization of polynomials involves expre 10 min read What is Factoring Trinomials Formula?A Trinomial is a polynomial with three terms. Examples of Trinomial are x+y+z, x2+2x+2, x+y-1 etc. A Trinomial can be of two types. They are Perfect Square Trinomial and Non-Perfect Square Trinomial. Factoring a polynomial is nothing but writing the expression polynomial as a product of two or more 4 min read a2 - b2 Formulaa2 - b2 formula in Algebra is the basic formula in mathematics used to solve various algebraic problems. a2 - b2 formula is also called the difference of squares formula, as this formula helps us to find the difference between two squares without actually calculating the squares. The image added bel 6 min read Difference of CubesDifference of Cubes is the formula in mathematics that is used to simplify the difference between two cubes. This formula is used to solve the difference of cubes without actually finding the cubes. This formula factorizes the difference of a cube and changes it into other forms. The difference of c 6 min read Discriminant Formula in Quadratic EquationsAlgebra can be defined as the branch of mathematics which deals with the study, alteration, and analysis of various mathematical symbols. It is the study of unknown quantities, which are often depicted with the help of variables in mathematics. Algebra has a plethora of formulas and identities for t 5 min read Sum of Arithmetic Sequence FormulaAn arithmetic sequence is a number series in which each subsequent term is the sum of its preceding term and a constant integer. This constant number is referred to as the common difference. As a result, the differences between every two successive terms in an arithmetic series are the same.If the f 5 min read Function Notation FormulaA function is a type of operator that takes an input variable and provides a result. When one quantity is dependent on another, a function is created. An interesting property of functions is that each input corresponds to a single output. In other words, such an operator between two sets, say set A 4 min read Binomial Distribution in ProbabilityBinomial Distribution is a probability distribution used to model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. This distribution is useful for calculating the probability of a specific number of successes in sce 14 min read Binomial Expansion FormulasBinomial expansion formula is a formula that is used to solve binomial expressions. A binomial is an algebraic expression with two terms. For example, x + y, x - a, etc are binomials. In this article, we have covered the Binomial Expansion definition, formulas, and others in detail.Table of ContentB 8 min read Binomial TheoremBinomial theorem is a fundamental principle in algebra that describes the algebraic expansion of powers of a binomial. According to this theorem, the expression (a + b)n where a and b are any numbers and n is a non-negative integer. It can be expanded into the sum of terms involving powers of a and 15+ min read FOIL MethodFOIL formula is used to perform multiplication between two binomials. A binomial is a polynomial with only two terms. Example: x+3, x2+4, 5x2+3x. A binomial is a 2 term algebraic expression that includes constants, variables, exponents and coefficients. Each letter in the FOIL represent steps to mul 2 min read Exponential Decay FormulaExponential Decay Formula: A quantity is said to be in exponential decay if it decreases at a rate proportional to its current value. In exponential decay, a quantity drops slowly at first before rapidly decreasing. The exponential decay formula is used to calculate population decay (depreciation), 7 min read Factorial FormulaThe factorial is one of the most fundamental mathematical operations in combinatorics, algebra, and number theory. Represented by an exclamation mark (!), the factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. It plays a crucial rol 8 min read Combinations Formula with ExamplesCombinations are way of selecting items from a collection of items. Different groups that can be formed by choosing r things from a given set of n different things, ignoring their order of arrangement, are called combinations of n things taken r at a time.The number of all such combinations is calcu 6 min read Fourier Series FormulaFourier Series is a sum of sine and cosine waves that represents a periodic function. Each wave in the sum, or harmonic, has a frequency that is an integral multiple of the periodic function’s fundamental frequency. Even though a Fourier series can include infinitely many harmonics, using just a few 15 min read Maclaurin seriesPrerequisite - Taylor theorem and Taylor series We know that formula for expansion of Taylor series is written as: f(x)=f(a)+\sum_{n=1}^{\infty}\frac{f^n(a)}{n!}(x-a)^n Now if we put a=0 in this formula we will get the formula for expansion of Maclaurin series. T hus Maclaurin series expansion can b 2 min read Coordinate GeometryMid Point Formula in Coordinate GeometryMid point formula in coordinate geometry provides a way to find the mid point of a line segment when the coordinates of the starting and ending points ( i.e. (x1, y1) and (x2, y2) )of the line segment is known. The mid point divides the line in two equal halves i.e the ratio of the sections of the l 6 min read Equation of a Straight Line | Forms, Examples and Practice QuestionsThe equation of a line describes the relationship between the x-coordinates and y-coordinates of all points that lie on the line. It provides a way to mathematically represent that straight path.In general, the equation of a straight line can be written in several forms, depending on the information 10 min read Equation of a CircleA circle is a geometric shape described as the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius. Some key components of the circle are:Center: The fixed point in the middle of the circ 14 min read Ellipse FormulaAn ellipse is a set of points such that the sum of the distances from any point on the ellipse to two fixed points (foci) is constant. In this article, we will learn about the ellipse definition, Ellipse formulas, and others in detail.Table of ContentWhat is Ellipse?What is Ellipse Formula?Major and 8 min read Trigonometry30-60-90 FormulaA 30-60-90 triangle is a special type of right triangle with one angle measuring 30°, another 60°, and the third angle (the right angle) measuring 90°. The 30-60-90 triangle is called a special right triangle as the angles of this triangle are in a unique ratio of 1:2:3. Here, a right triangle means 7 min read Cofunction FormulasA trigonometric cofunction is defined as expressing a trigonometric angle ratio in terms of the other. It illustrates how sine, cosine, tangent, cotangent, secant, and cosecant relate to each other. The cofunction of an angle's complement is equal to that angle's trigonometric function. For example, 7 min read What is Cos Square theta Formula?The equations that relate the different trigonometric functions for any variable are known as trigonometric identities. These trigonometric identities help us to relate various trigonometric formulas and relationships with different angles. They are sine, cosine, tangent, cotangent, sec, and cosec. 3 min read What are Cosine Formulas?Trigonometry is a discipline of mathematics that studies the relationships between the lengths of the sides and angles of a right-angled triangle. Trigonometric functions, also known as goniometric functions, angle functions, or circular functions, are functions that establish the relationship betwe 8 min read Cosecant FormulaCosecant is one of the six basic trigonometric ratios and its formula is cosecant(θ) = hypotenuse/opposite, it is also represented as, csc(θ). It is the inverse(reciprocal) ratio of the sine function and is the ratio of the Hypotenus and Opposite sides in a right-angle triangle. In this article, we 4 min read Cotangent FormulaTrigonometry is an important branch of mathematics that deals with the relation between the lengths of sides and angles of a right-angled triangle. Sine, Cosine, tangent, cosecant, secant, and cotangent are the six trigonometric ratios or functions. Where a trigonometric ratio is depicted as the rat 7 min read Tangent FormulasTangent Function is among the six basic trigonometric functions and is calculated by taking the ratio of the perpendicular side and the hypotenuse side of the right-angle triangle.In this article, we will learn about Trigonometric ratios, Tangent formulas, related examples, and others in detail.Tabl 8 min read Cot Half Angle FormulaTrigonometry is a branch of mathematics that uses trigonometric ratios to determine the angles and incomplete sides of a triangle. The trigonometric ratios such as sine, cosine, tangent, cotangent, secant, and cosecant are used to investigate this branch of mathematics. It's the study of how the sid 6 min read 2cosA cosB FormulaThe identity 2 cos A cos B = cos(A + B) + cos(A – B) is one of the important product-to-sum formulas in trigonometry. This identity is used to convert a product of cosine functions into a sum of cosines, which can simplify the process of solving trigonometric equations, evaluating integrals, and per 7 min read Multiple Angle FormulasTrigonometry is one of the important topics in mathematics that is used in various fields. The trigonometric formulae are applied and used in various formulae, derivations, etc. This article is about the multiple angle formulae in trigonometry where we find sine, cosine, and tangent for multiple ang 5 min read Double Angle Formula for CosineDouble angle formula for cosine is a trigonometric identity that expresses cosâ¡(2θ) in terms of cosâ¡(θ) and sinâ¡(θ) the double angle formula for cosine is, cos 2θ = cos2θ - sin2θ. The formula is particularly useful in simplifying trigonometric expressions and solving equations involving trigonometri 5 min read Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function 11 min read Complex NumberComplex Number FormulaThe sum of a real number and an imaginary number is defined as a complex number, and the numbers that are not real numbers are called imaginary numbers. The number can be written in the form of b+ic, where b and c are real numbers ic is an imaginary number, and †i†is an imaginary part which is cal 13 min read Absolute Value of a Complex NumberThe absolute value (also called the modulus) of a complex number z = a + bi is its distance from the origin in the complex plane. The absolute value tells you how far a number is from zero, regardless of its direction (positive or negative).It is denoted as ∣z∣ and is given by the formula:|z| = \sqr 7 min read Complex Number Power FormulaComplex Numbers are numbers that can be written as a + ib, where a and b are real numbers and i (iota) is the imaginary component and its value is √(-1), and are often represented in rectangle or standard form. 10 + 5i, for example, is a complex number in which 10 represents the real component and 5 6 min read DeMoivre's TheoremDe Moivre's theorem is one of the fundamental theorem of complex numbers which is used to solve various problems of complex numbers. This theorem is also widely used for solving trigonometric functions of multiple angles. DeMoivre’s Theorem is also called “De Moivre’s Identity†and “De Moivre’s Form 6 min read Covariance MatrixA Covariance Matrix is a type of matrix used to describe the covariance values between two items in a random vector. It is also known as the variance-covariance matrix because the variance of each element is represented along the matrix’s major diagonal and the covariance is represented among the no 10 min read Determinant of Matrix with Solved ExamplesThe determinant of a matrix is a scalar value that can be calculated for a square matrix (a matrix with the same number of rows and columns). It serves as a scaling factor that is used for the transformation of a matrix.It is a single numerical value that plays a key role in various matrix operation 15+ min read CalculusLimit FormulaLimits help us comprehend how functions behave as their inputs approach certain values. Think of a limit as the destination that a function aims to reach as the input gets closer and closer to a specific point.In this article, we will explore the essential limit formulas that form the backbone of ca 7 min read Average and Instantaneous Rate of ChangeThe average rate of change represents the total change in one variable in relation to the total change of another variable. Instantaneous rate of change, or derivative, measures the specific rate of change of one variable in relation to a specific, infinitesimally small change in the other variable. 8 min read Calculus | Differential and Integral CalculusCalculus was founded by Newton and Leibniz. Calculus is a branch of mathematics that helps us study change. It is used to understand how things change over time or how quantities grow, shrink, or accumulate. There are two main parts of calculus:Differential Calculus: It helps us calculate the rate o 4 min read Total DerivativeTotal Derivative of a function measures how that function changes as all of its input variables change. For function f at a point is an approximation near the point of the function w.r.t. (with respect to) its arguments (variables).It is an approximation of the actual change in the function and is u 5 min read Difference Quotient FormulaThe Difference Quotient Formula is a part of the definition of a function derivative. One can get derivative of a function by applying Limit h tends to zero i.e., h ⇢ 0 on difference quotient function. The difference quotient formula gives the slope of the secant line. A secant line is a line that p 5 min read Chain Rule: Theorem, Formula and Solved ExamplesThe Chain Rule is a way to find the derivative of composite functions. It is one of the basic rules used in mathematics for solving differential equations. It helps us to find the derivative of composite functions such as (3x2 + 1)4, (sin 4x), e3x, (ln x)2, and others. Only the derivatives of compos 8 min read Implicit DifferentiationImplicit Differentiation is the process of differentiation in which we differentiate the implicit function without converting it into an explicit function. For example, we need to find the slope of a circle with an origin at 0 and a radius r. Its equation is given as x2 + y2 = r2. Now, to find the s 5 min read Antiderivative: Integration as Inverse Process of DifferentiationAn antiderivative is a function that reverses the process of differentiation. It is also known as the indefinite integral. If F(x) is the antiderivative of f(x), it means that:d/dx[F(x)] = f(x)In other words, F(x) is a function whose derivative is f(x).Antiderivatives include a family of functions t 7 min read Integration FormulasIntegration Formulas are the basic formulas used to solve various integral problems. They are used to find the integration of algebraic expressions, trigonometric ratios, inverse trigonometric functions, and logarithmic and exponential functions. These integration formulas are beneficial for finding 10 min read Integration by PartsIntegration by Parts or Partial Integration, is a technique used in calculus to evaluate the integral of a product of two functions. The formula for partial integration is given by:∫ u dv = uv - ∫ v duWhere u and v are differentiable functions of x. This formula allows us to simplify the integral of 9 min read Integration by Substitution FormulaThe process of finding the anti-derivative of a function is the inverse process of differentiation i.e. finding integral is the inverse process of differentiation. Integration can be used to find the area or volume of a function with or without certain limits or boundaries It is shown as∫g(x)dx = G( 5 min read Definite Integral | Definition, Formula & How to CalculateA definite integral is an integral that calculates a fixed value for the area under a curve between two specified limits. The resulting value represents the sum of all infinitesimal quantities within these boundaries. i.e. if we integrate any function within a fixed interval it is called a Definite 8 min read Area Under CurveArea Under Curve is area enclosed by curve and the coordinate axes, it is calculated by taking very small rectangles and then taking their sum if we take infinitely small rectangles then their sum is calculated by taking the limit of the function so formed.For a given function f(x) defined in the in 11 min read Differentiation and Integration FormulaDifferentiation and Integration are two mathematical operations used to find change in a function or a quantity with respect to another quantity instantaneously and over a period, respectively. Differentiation is an instantaneous rate of change, and it breaks down the function for that instant with 11 min read Differential EquationsA differential equation is a mathematical equation that relates a function with its derivatives. Differential Equations come into play in a variety of applications such as Physics, Chemistry, Biology, Economics, etc. Differential equations allow us to predict the future behavior of systems by captur 12 min read Like