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 nPr.
nPr = 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 nCr.
nCr = 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.