Factorial Formula

Factorial of a number ‘n’ is defined as the product of all the whole numbers less than ‘n’ up to 1. So, it can be defined as a factorial for a number 4 as 4 × 3 × 2 × 1 = 24. It is represented by the symbol ‘!’. Suppose, the factorial of 5 is needed to be written, it can be written as 5! and the value of 5! is 5 × 4 × 3 × 2 × 1 = 120. Let’s take a look at the factorial formula in generalized form, 

Factorial Formula

As discussed earlier, the factorial of a certain number is the multiplication of that number with all the numbers lesser than that number up to 1. So, if the number is n, and that factorial of n is needed to be found, n should be multiplied with (n – 1), (n – 2),… up to 1. The formula for factorial will become,

Factorial of n = n! = n × (n – 1) × (n – 2) × … × 1

Properties of Factorial

• Factorial of any number is a whole number
• A factorial can also be represented as a recursive function.

n! = n × (n – 1) × (n – 2) × … × 1 = n × (n – 1)!

• Factorial of zero is 1, that is 0! = 1
• Factorial of negative numbers is not defined

Uses of the factorial formula

The factorial formula is used in many areas, specifically in permutations and combinations of mathematics. For example,

• The number of ways n distinct objects can be arranged in a row is equal to n!
• Permutation gives the number of ways to select r elements from n elements when order matters. It is given using the formula ⁿP_r.

ⁿP_r = n! / (n – r)!

• Combination gives the number of ways to select r elements from n elements where order does not matter. It is given as ⁿC_r.

ⁿC_r = n! / r! (n – r)!

Sample Problems

Question 1: Find the value of factorial of 5.

Solution:

To find the factorial of 5, we need to multiply all the whole numbers smaller than or equal to 5.

5! = 5 × 4 × 3 × 2 × 1 = 120

Hence, 5! = 120

Question 2: Find the value of a number x, given factorial of x, is 720.

Solution:

Apply the recursive property of factorial to find x. Until and unless we get 720 as our result, we will proceed recursively.

1! = 1

2! = 2 × 1! = 2

3! = 3 × 2! =6

4! = 4 × 3! = 4 × 6 = 24

5! = 5 × 4! = 5 × 24 = 120

6! = 6 × 5! = 6 × 120 = 720

Since 720 is obtained as the factorial of 6, one can compare the value of x with 6.

Thus, the value of x = 6

Question 3: Find the number of ways 5 distinct objects can be arranged in a row.

Solution:

Use the property that the number of ways n distinct objects can be arranged in a row is equal to n!

Thus, 5 distinct objects can be arranged in 5! = 5 × 4 × 3 × 2 × 1 = 120.

So, the number of ways is equal to 120.

Question 4: Find the number of ways 3 students can be selected from a class of 50 students.

Solution:

To find the number of ways 3 students can be selected from a class of 50 students, we can use the formula for Combination, since the order of the selected three students does not matter here.

Thus, the total number of ways = ⁵⁰C₃

So, this can be simplified as ⁵⁰C₃ = 50! / (3! × 47!) = (50 × 49 × 48 × 47!) / (3! × 47!) = 50 ×49 × 48 / 6 = 19,600

So, there are a total of 19,600 ways.

Question 5: Three different fruits are to be distributed among a group of 10 people. Find the total number of ways this can be possible.

Solution:

Since, in this case, the order of how the fruits are distributed matters, we need to implement Permutation.

So, the total number of ways is given as ¹⁰P₃.

Simplifying, this can be written as,

¹⁰P₃ = 10! / (10 – 3) ! = 10! / 7! = 10 × 9 × 8 × 7! / 7! = 10 × 9 × 8 = 720

Thus, there are a total of 720 ways possible.