Solved Examples on Factorial
Example 1: Evaluate the following.
- Factorial of 1
- Factorial of 3
- Factorial of 4
- Factorial of 6
- Factorial of 7
- Factorial of 8
- Factorial of 9
Solution:
Factorial of 1 = 1! = 1
Factorial of 3 = 3! = 3 Γ 2 Γ 1 = 6
Factorial of 4 = 4! = 4 Γ 3 Γ 2 Γ 1 = 24
Factorial of 6 = 6! = 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 = 720
Factorial of 7 = 7! = 7 Γ 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 = 5040
Factorial of 8 = 8! = 8 Γ 7 Γ 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 = 40320
Factorial of 9 = 9! = 9 Γ 8 Γ 7 Γ 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 =362880
Example 2: What is the value of factorial: 14! / (11! Γ 4!)
Solution:
14! / (11! Γ 4!) = (14 Γ 13 Γ 12 Γ 11!) / (11! Γ 4!)
β 14! / (11! Γ 4!) = (14 Γ 13 Γ 12) / 4!
β 14! / (11! Γ 4!) = (14 Γ 13 Γ 12) / (4 Γ 3 Γ 2 Γ 1!)
β 14! / (11! Γ 4!) = (14 Γ 13 Γ 12) / (12 Γ 2 )
β 14! / (11! Γ 4!) = (7 Γ 13)
β 14! / (11! Γ 4!) = 91
Example 3: Evaluate the expression 6! β 3!
Solution:
6! β 3! = (6 Γ 5 Γ 4 Γ 3!) β 3!
β 6! β 3! = (6 Γ 5 Γ 4 Γ 3!) β 3!
β 6! β 3! = (120 Γ 3!) β 3!
β 6! β 3! = 3![120 β 1]
β 6! β 3! = 6 Γ 119
β 6! β 3! = 714
Example 4: If (1 / 6!) = (x / 8!) β (1 / 7!), then find the value of x.
Solution:
(1 / 6!) = (x / 8!) β (1 / 7!)
β (1 / 6!) = (x / 8 Γ 7!) β (1 / 7!)
β (1 / 6!) = (1 / 7!)[(x / 8) β 1]
β (1 / 6!) = {1 / (7 Γ6!)}[(x / 8) β 1]
β (1 / 6!) = (1 / 6!)(1 / 7 )[(x / 8) β 1]
β 1 = (1 / 7 )[(x / 8) β 1]
β 7 = (x / 8) β 1
β (x / 8) = 7 + 1
β (x / 8) = 8
β x = 64
Example 5: How many 4-digit numbers can be formed using the digits 4,6,7,9 in each of which no digit is repeated?
Solution:
Given:
Digits: 4, 6, 7, and 9
Number of digits = 4
We have to arrange these digits to form a 4-digit number.
The number of ways for arranging these digits to form a 4-digit number is 4!
and 4! = 4 Γ 3 Γ 2 Γ 1 = 24
Thus, there are 24 ways in which a 4 digit number can be formed without repeating the digits.
Example 6: Evaluate the expression 3! (2! Γ 0!)
Solution:
3! (2! Γ 0!) = (3 Γ 2 Γ 1) (2 Γ 1 Γ 1) [By using factorial formula and 0! = 1]
β 3! (2! Γ 0!) = 6 Γ 2
β 3! (2! Γ 0!) = 12
Factorial
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 βnβ is defined as the product of all natural numbers till n.
In this article, weβll delve into the intricacies of factorials, exploring factorial notation, the diverse range of factorial formulas, and techniques for computing factorials. Additionally, weβll touch upon the properties and practical applications of factorials, provide illustrative examples, and address common questions pertaining to this topic. Letβs embark on our journey of understanding factorials.
Table of Content
- What is Factorial?
- Factorial Formula
- How to Find Factorial of a Number?
- Factorial Examples
- Properties of Factorial
- Factorials 1 to 20
- Applications of Factorials
- Solved Examples on Factorial