SIMPLIFY THE FACTORIAL EXPRESSION
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
Determine the value for each expression. Simplify fully before using a calculator.
Problem 1 :
8! / 5!
Solution :
8! / 5! = (8 × 7 × 6 × 5!) / 5!
= 8 × 7 × 6
8! / 5! = 336
Problem 2 :
19! / 13!
Solution :
19! / 13! = (19 × 18 × 17 × 16 × 15 × 14 × 13!) / 13!
= 19 × 18 × 17 × 16 × 15 × 14
19! / 13! = 19535040
Problem 3 :
21! / 17! 4!
Solution :
21! / 17! 4! = (21 × 20 × 19 × 18 × 17!) / 17! (4 × 3 × 2 × 1)
= (21 × 20 × 19 × 18) / (4 × 3 × 2 × 1)
= 143640 / 24
21! / 17! 4! = 5985
Problem 4 :
9! / 7! 2!
Solution :
9! / 7! 2! = (9 × 8 × 7!) / 7! (2 × 1)
= 72 / 2
9! / 7! 2! = 36
Problem 5 :
155! / 152!
Solution :
155! / 152! = (155 × 154 × 153 × 152!) / 152!
= 155 × 154 × 153
155! / 152! = 3652110
Problem 6 :
93! / 89! 4!
Solution :
93! / 89! 4! = (93 × 92 × 91 × 90 × 89!) / 89! (4 × 3 × 2 × 1)
= (93 × 92 × 91 × 90) / (4 × 3 × 2 × 1)
= 70073640 / 24
93! / 89! 4! = 22919735
Determine the value for each expression. Simplify fully before using a calculator.
Problem 7 :
10! / 5!
Solution :
10! / 5! = (10 × 9 × 8 × 7 × 6 × 5!) / 5!
= 10 × 9 × 8 × 7 × 6
10! / 5! = 30240
Problem 8 :
21! / 14!
Solution :
21! / 14! = (21 × 20 × 19 × 18 × 17 × 16 × 15 × 14!) / 14!
= 21 × 20 × 19 × 18 × 17 × 16 × 15
21! / 14! = 586051200
Problem 9 :
9! / 3! 6!
Solution :
9! / 3! 6! = (9 × 8 × 7 × 6!) / 6! (3 × 2 × 1)
= (9 × 8 × 7) / (3 × 2 × 1)
= 504 / 6
9! / 3! 6! = 84
Problem 10 :
12! / 8! 4!
Solution :
12! / 8! 4! = (12 × 11 × 10 × 9 × 8!) / 8! (4 × 3 × 2 ×1)
= (12 × 11 × 10 × 9) / (4 × 3 × 2 ×1)
= 11880 / 24
12! / 8! 4! = 495
Problem 11 :
7! /2! 5! + 7! / 4! 3!
Solution :
7! /2! 5! + 7! / 4! 3! = (7 × 6 × 5!) / 5! (2 × 1) + (7 × 6 × 5 × 4!) / 4! (3 × 2 × 1)
= (42 / 2) + (210 / 6)
= 21 + 35
7! /2! 5! + 7! / 4! 3! = 56
Problem 12 :
(15! / 9! 6!) + (15! / 10! 5!)
Solution :
15! / 9! 6! = (15 × 14 × 13 × 12 × 11 × 10 × 9!) / 9! (6 × 5 × 4 × 3 × 2 ×1)
= (15 × 14 × 13 × 12 × 11 × 10) / (6 × 5 × 4 × 3 × 2 ×1)
= 3603600 / 720
15! / 9! 6! = 5005
15! / 10! 5! = (15 × 14 × 13 × 12 × 11 × 10!) / 10! (5 × 4 × 3 × 2 ×1)
= (15 × 14 × 13 × 12 × 11) / (5 × 4 × 3 × 2 ×1)
= 360360 / 120
15! / 10! 5! = 3003
15! / 9! 6! + 15! / 10! 5! = 5005 + 3003
15! / 9! 6! + 15! / 10! 5! = 8008
Problem 13 :
2 × 5! / 2! 3!
Solution :
2 × 5! / 2! 3! = 2 × (5 × 4 × 3!) / 3! (2 × 1)
= 2 × (5 × 4) / 2
2 × 5! / 2! 3! = 20
Problem 14 :
3 × 11! / 7! 4!
Solution :
3 × 11! / 7! 4! = 3 × (11 × 10 × 9 × 8 × 7!) / 7! (4 × 3 × 2 × 1)
= 3 × (11 × 10 × 9 × 8) / (4 × 3 × 2 × 1)
= 3 × 330
3 × 11! / 7! 4! = 990
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