SIMPLIFYING FACTORIAL EXPRESSIONS
What is factorial ?
Factorial of a whole number n is defined as the product of that number with every whole number less than or equal to n till 1.
For all positive integers, n! is defined as
n! = n (n−1) (n−2)⋯(2)(1)
4! = 4 x 3 x 2 x 1
4! = 24
How to simplify factorial expression ?
To simplify factorial expression, first we have to choose the larger value and write it in descending order as product of terms.
Simplify without using a calculator.
Problem 1 :
6! / 5!
Solution :
6! / 5! = (6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1) / (5 ∙ 4 ∙ 3 ∙ 2 ∙ 1)
= 6
Alternatively :
6! = (6 ∙ 5!) / 5!
= 6
Simplify without using a calculator.
Problem 2 :
6! / 4!
Solution :
6! / 4! = (6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1) / (4 ∙ 3 ∙ 2 ∙ 1)
= 6 ∙ 5
= 30
Alternatively :
6! = (6 ∙ 5 ∙ 4!) / 4!
= 6 ∙ 5
= 30
Problem 3 :
6! / 7!
Solution :
6! / 7! = (6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1) / (7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1)
= 1/7
Alternatively :
6! / 7! = 6! / (7 ∙ 6!)
= 1/7
Problem 4 :
4! / 6!
Solution :
4! / 6! = (4 ∙ 3 ∙ 2 ∙ 1) / (6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1)!
= 1/30
Problem 5 :
100! / 99!
Solution :
100! / 99! = 100 ∙ 99! / 99!
= 100
Problem 6 :
7! / (5! × 2!)
Solution :
7! / 5! × 2! = 7 ∙ 6 ∙ 5! / 5! ∙ (2 ∙ 1!)
= 42/2
= 21
Problem 7 :
Simplify:
n! / (n - 1)!
Solution :
= n (n - 1)! / (n - 1)!
= n
Problem 8 :
(n + 2)! / n!
Solution :
= [(n + 2) (n + 1) n!] / n!
= (n + 2) (n + 1)
Problem 9 :
(n + 1)! / (n - 1)!
Solution :
= (n + 1) n (n - 1)! / (n - 1)!
= n(n + 1)
Simplify the following expressions.
Problem 10 :
5! / 6!
Solution :
= (5 ∙ 4 ∙ 3 ∙ 2 ∙ 1) / (6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1)
= 1/6
Problem 11 :
7! / 5!
Solution:
= (7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1) / (5 ∙ 4 ∙ 3 ∙ 2 ∙ 1)
= 7 ∙ 6
= 42
Problem 12 :
(3!)² / 9
Solution :
= (3 ∙ 2 ∙ 1!)² / 9
= 6² / 9
= 36 / 9
= 4
Problem 13 :
(3! ∙ 4!) / 5!
Solution :
= (3 ∙ 2 ∙ 1) ∙ 4! / 5 ∙ 4!
= 3 ∙ 2 ∙ 1 / 5
= 6/5
Problem 14 :
9! /(5! ∙ 3!)
Solution :
= [9 ∙ 8 ∙ 7 ∙ 6 ∙ 5!] / [5! ∙ (3 ∙ 2 ∙ 1)]
= (9 ∙ 8 ∙ 7 ∙ 6) / (3 ∙ 2 ∙ 1)
= 3024 / 6
= 504
Problem 15 :
(4 - 1)! / 4!
Solution :
= 3! / 4!
= 3! / (4 ∙ 3!)
=1/4
Problem 16 :
(2 ∙ 3)! / 3!
Solution :
= 6! / 3!
= (6 ∙ 5 ∙ 4 ∙ 3!)) / 3!
= 6 ∙ 5 ∙ 4
= 120
Problem 17 :
88! / 90!
Solution :
= 88! / (90 ∙ 89 ∙ 88!)
= 1 / (90 ∙ 89)
= 1/8010
Problem 18 :
(77! ∙ 2!)/ 78!
Solution :
= 77! ∙ (2 ∙ 1) / (78 ∙ 77!)
= 2/78
= 1/39
Problem 19 :
38! ∙ 3! / 39!
Solution :
= 38! ∙ (3 ∙ 2 ∙ 1) / (39 ∙ 38!)
= 3 ∙ 2 ∙ 1 / 39
= 6/39
= 2/13
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