Applications of Factorials

There are various applications of the factorial. Some of the applications of factorials are listed below:

• Factorial is used in permutations.
• Factorials are used in combinations.
• It is used in probability formulas.
• It is used in binomial expansion.

Factorials in Combinatorics

In calculation of both permutation and combination is used as the formula for both involves the factorials. Let’s see Permutation Formula and Combination Formula along with their examples.

Permutation Formula

The formula for calculating Permutation, denoted as ⁿP_r  which represents the number of ways to arrange r objects from a set of n distinct objects without repetition and formula for permutation is given by:

ⁿP_r = n! / (n – r)!

Where,

• n is the total number of distinct objects to choose from,
• r is the number of objects to be chosen and arranged,
• n! is the product of all positive integers from 1 to n,
• (n – r)! is the product of all positive integers from 1 to (n – r).

Let us take an example for this:

Example: Evaluate the value of ⁵P₃.

Solution:

By permutation formula

ⁿP_r = n! / (n – r)!

⇒ ⁵P₃ = 5! / (5 – 3)!

⇒ ⁵P₃ = 5! / 2!

⇒ ⁵P₃ = 120 / 2

⇒ ⁵P₃ = 60

Combination Formula

The formula for calculating Combination, denoted as ⁿC_r, where n is the total number of items to choose from, and r is the number of items to choose without replacement. This formula is given as follows:

ⁿC_r = n! / [r! × (n – r)!]

Where,

• n is the total number of distinct objects to choose from,
• r is the number of objects to be chosen and arranged,
• n! is the product of all positive integers from 1 to n,
• (n – r)! is the product of all positive integers from 1 to (n – r).

Let us take an example for this:

Example: Find the value of ⁴C₂.

Solution:

By combination formula

ⁿC_r = n! / [r! × (n – r)!]

⇒ ⁴C₂ = 4! / [2! × (4 – 2)!]

⇒ ⁴C₂ = 4! / [2! × 2!]

⇒ ⁴C₂ = 24 / [2 × 2]

⇒ ⁴C₂ = 24 / 4

⇒ ⁴C₂ = 12

Factorials in Probability

Factorials are used in multiple formulas in probability, as factorials help us calculate the number of ways of things with the help of principle of counting, permutation, and combination. Let’s consider an example of Probability where we calculate the probability of any event with the help of factorials.

Example: A box contains different colored balls. There is 15% chance of getting a red ball. What is the probability that exactly 4 balls are red out of 10.

Solution:

Applying binomial distribution

P(X = r) = ⁿC_r p^r q^n-r

n = 10, p = 0.15, q = 0.85, r = 4

⇒ P(X = 4) = ¹⁰C₄ (0.15)⁴ (0.85)⁶

⇒ P(X = 4) = [10! / {4! × 6!}] (0.15)⁴ (0.85)⁶

⇒ P(X =4) = [{10× 9 × 8 × 7} / 24] (0.15)⁴ (0.85)⁶

⇒ P(X = 4) = 0.04

Also, Check

• Number System
• Binomial Theorem
• Permutation and Combination

Factorial is a fundamental concept in combinatorics as factorials play important roles in various mathematical formulas such as permutations, combinations, probability, and many other formulas. Factorial of any natural number “n” is defined as the product of all natural numbers till n.

In this article, we’ll delve into the intricacies of factorials, exploring factorial notation, the diverse range of factorial formulas, and techniques for computing factorials. Additionally, we’ll touch upon the properties and practical applications of factorials, provide illustrative examples, and address common questions pertaining to this topic. Let’s embark on our journey of understanding factorials.