Mathematics | PnC and Binomial Coefficients

Permutations and Combinations (PnC) are fundamental concepts in combinatorics and are essential for counting and arranging objects. Binomial coefficients are closely related and used in various mathematical formulas, including the Binomial Theorem. These concepts are widely used in engineering, computer science, probability, and statistics for solving problems involving discrete structures and optimization.

Read More: Combinatorics Basics

What are Permutations?

A permutation is an arrangement of objects in a specific order. The number of permutations of n distinct objects taken r at a time is given by:

P(n,r) =

Example: Number of ways to arrange 3 out of 5 objects:

P(5,3) = 5! / (5−3)! = 5!/2! = 60.

Read More: Permutation - Formula, Definition, Examples

What are Combinations?

A combination is a selection of objects without regard to the order. The number of combinations of n distinct objects taken r at a time is given by:

C(n, r) =

Example: Number of ways to choose 3 out of 5 objects:

C(5,3) = 5! / 3!2! = 10.

Read More: Combinations

Binomial Coefficients

(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k

Binomial Coefficient Example

Expand: (x + y)³

(x + y)³ = C(3,0)x³y⁰ + C(3,1)x²y¹ + C(3,2)x¹y² + C(3,3)x⁰y³

= 1x³ + 3x²y + 3xy² + 1y³

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Solved Examples on PnC and Binomial Coefficients

Permutation

Problem 1: How many ways can 5 books be arranged on a shelf?

Solution: This is a straightforward permutation.

P(5) = 5! = 5 × 4 × 3 × 2 × 1 = 120 ways

Combination

Problem 2: In how many ways can a committee of 3 be chosen from 10 people?

Solution: This is a combination problem.

C(10,3) = 10! / (3! × 7!) = 120

Binomial Coefficient

Problem 3 : Find the coefficient of x³y² in the expansion of (x + y)⁵

Solution: This is the binomial coefficient ⁵C₃ (as we need to choose 3 x's out of 5 terms)

⁵C₃ = 5! / (3! × 2!) = 10

Permutation with Repetition

Problem 4: How many 4-digit numbers can be formed using the digits 1, 2, 3 if repetition is allowed?

Solution: We have 3 choices for each of the 4 positions.

3⁴ = 81 numbers

Combination with Repetition

Problem 5: How many ways are there to select 4 ice cream scoops if there are 5 flavors available?

Solution: This is a combination with repetition.

C(5+4-1, 4) = C(8,4) = 8! / (4! × 4!) = 70 ways

Binomial Theorem Application

Problem 6: Find the value of (1 + x)¹⁰ when x = 0.1

Solution: Using the binomial theorem:

(1 + 0.1)¹⁰ = ¹⁰C₀ + ¹⁰C₁(0.1) + ¹⁰C₂(0.1)² + ... + ¹⁰C₁₀(0.1)¹⁰

≈ 2.5937 (rounded to 4 decimal places)

Permutation with Restrictions

Problem 7: How many ways can 8 people be seated in a row if 2 specific people must sit together?

Solution: Treat the 2 people as 1 unit. So we have 7 units to arrange.

7! × 2! = 5040 × 2 = 10,080 ways

Combination Problem

Problem 8: From a standard 52-card deck, how many 5-card hands contain exactly 2 hearts?

Solution: Choose 2 hearts from 13 hearts, and 3 non-hearts from 39 non-hearts.

C(13,2) × C(39,3) = 78 × 9139 = 712,842 hands

Binomial Probability

Problem 9: If the probability of a defective item is 0.1, what's the probability of exactly 2 defectives in a sample of 5?

Solution: Use binomial probability formula:

P(X=2) = ⁵C₂ (0.1)² (0.9)³ = 10 × 0.01 × 0.729 = 0.0729 or about 7.29%

Permutation with Indistinguishable Objects:

Problem 10 : How many distinct ways can the letters of "MISSISSIPPI" be arranged?

Solution: 11! / (4! × 4! × 2!) = 34,650 ways

Applications in Engineering

1. Probability and Statistics

Permutations and combinations are used to calculate probabilities and analyze statistical data.

2. Computer Science

In algorithms and data structures, permutations and combinations are used for optimization and enumeration problems.

3. Operations Research

Permutations and combinations are used in resource allocation, scheduling, and optimization problems.

4. Cryptography

Permutations and combinations are used in encryption algorithms to create secure keys and ensure data integrity.

5. Network Design

In designing efficient networks, combinations are used to determine the optimal number of connections.

Conclusion - PnC and Binomial Coefficients

Permutations and combinations, along with binomial coefficients, are essential tools in combinatorics with wide-ranging applications in engineering, computer science, and other fields. Understanding these concepts allows for efficient problem-solving and optimization in various domains.

GATE CS Corner Questions on PnC and Binomial Coefficients

Practicing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests. It is highly recommended that you practice them. 

1. GATE CS 2007, Question 84 
2. GATE CS 2007, Question 85 
3. GATE CS 2003, Question 4 
4. GATE CS 2003, Question 5