Combinatorics

Combinatorics is the branch of Mathematics dealing with the study of finite or countable discrete structures. It includes the enumeration or counting of objects having certain properties. Counting helps us solve several types of problems such as counting the number of available IPv4 or IPv6 addresses. 

Permutation and Combination are fundamental concepts in combinatorics that help us solve problems involving arrangements and selections. While permutation deals with arranging objects in a specific order, combination focuses on selecting objects without considering the order.

Permutation and Combination for Students and Beginners

This section introduces basic concepts like factorial, permutations, and combinations with easy explanations for new learners.

• Factorial
• Fundamental Counting Principle
• Permutation 
• Permutation Formulas
• Combination 
• Combination Formulas
• Difference between Permutations and Combinations
• Principle of Inclusion and Exclusion
• Real-Life Applications of Permutations and Combinations

Permutation and Combination for Aptitude Preparation

Practice aptitude-style problems and quizzes to prepare for competitive exams using permutations and combinations.

• Permutation and Combination – Aptitude Questions and Answers
• Problem on permutations and combinations – set 1 | set 2
• Problems On Permutations | Set 1
• Permutation and Combination- Quiz

Advanced Topics of Permutation and Combination

Explore deeper topics like permutation groups, their properties, and related theorems in advanced combinatorics.

• Permutation Groups and Multiplication of Permutation
• Inverse of Permutation Group
• Order of Permutation Group
• Even and Odd Permutations and their theorems

Permutation and Combination for Programmers

This section covers coding problems based on permutations and combinations, helping you implement logic and solve real-world programming challenges.

• Program to calculate value of nCr using Recursion
• Permutations of given String
• All Permutations of an Array
• Ways to sum to N using Natural Numbers up to K with repetitions allowed
• Count all possible N-length vowel permutations
• Generate a combination of minimum coins
• Number of ways to go from one point to another in a grid
• Different ways to represent N as sum of K non-zero integers
• Number of arrays of size N whose elements are positive integers and sum is K
• Generate all cyclic permutations of a number

What do you mean by permutations and combinations?

A permutation is an act of arranging things in a specific order, represented as ( ⁿp_r ). Combinations are the ways of selecting r objects from a group of n objects, where the order of the object chosen does not affect the total combination and is represented as ( ⁿC_r )..

How are permutations different from combinations?

Permutations consider the order of objects, while combinations do not.

What is the Rule of Product in combinatorics?

The Rule of Product states that if there are n ways to do one thing and mmm ways to do another, then there are n×m ways to do both.

What is the factorial formula?

Factorial formula is used for the calculation of permutations and combinations. The factorial formula for n! is given as

n! = n × (n-1) × . . . × 4 × 3 × 2 × 1

For example, 3! = 3 × 2 × 1 = 6 and 5! = 5 × 4 × 3 × 2 × 1 = 120.

What does ⁿC_r represent?

ⁿC_r represents the number of combinations that can be made from “n” objects taking “r” at a time.

Write examples of permutations and combinations.

Number of 3-letter words that can be formed by using the letters of the word says, HELLO; ⁵P₃ = 5!/(5-3)! this is an example of a permutation.
Number of combinations we can write the words using the vowels of the word HELLO; ⁵C₂ =5!/[2! (5-2)!], this is an example of a combination.

How is combinatorics used in cryptography?

Combinatorics is used to generate cryptographic keys and analyze cryptographic algorithms to ensure data security.

Why is the binomial theorem important in combinatorics?

The binomial theorem provides a formula for expanding binomial expressions and involves binomial coefficients, which are fundamental in combinatorial calculations.

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Article Tags :

• Engineering Mathematics
• GATE CS
• Discrete Mathematics
• Pigeonhole Principle