SIMPLIFY THE FACTORIAL EXPRESSION
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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 (nā1) (nā2)āÆ(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 ā 5 ā 4 ā 3 ā 2 ā 1) / (5 ā 4 ā 3 ā 2 ā 1)
= 6
Alternatively :
6! = (6 ā 5!) / 5!
= 6
Determine the value for each expression. Simplify fully before using a calculator.
Problem 1 :
8! / 5!
Solution :
8! / 5! = (8 Ć 7 Ć 6 Ć 5!) / 5!
= 8 Ć 7 Ć 6
8! / 5! = 336
Problem 2 :
19! / 13!
Solution :
19! / 13! = (19 Ć 18 Ć 17 Ć 16 Ć 15 Ć 14 Ć 13!) / 13!
= 19 Ć 18 Ć 17 Ć 16 Ć 15 Ć 14
19! / 13! = 19535040
Problem 3 :
21! / 17! 4!
Solution :
21! / 17! 4! = (21 Ć 20 Ć 19 Ć 18 Ć 17!) / 17! (4 Ć 3 Ć 2 Ć 1)
= (21 Ć 20 Ć 19 Ć 18) / (4 Ć 3 Ć 2 Ć 1)
= 143640 / 24
21! / 17! 4! = 5985
Problem 4 :
9! / 7! 2!
Solution :
9! / 7! 2! = (9 Ć 8 Ć 7!) / 7! (2 Ć 1)
= 72 / 2
9! / 7! 2! = 36
Problem 5 :
155! / 152!
Solution :
155! / 152! = (155 Ć 154 Ć 153 Ć 152!) / 152!
= 155 Ć 154 Ć 153
155! / 152! = 3652110
Problem 6 :
93! / 89! 4!
Solution :
93! / 89! 4! = (93 Ć 92 Ć 91 Ć 90 Ć 89!) / 89! (4 Ć 3 Ć 2 Ć 1)
= (93 Ć 92 Ć 91 Ć 90) / (4 Ć 3 Ć 2 Ć 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 Ć 9 Ć 8 Ć 7 Ć 6 Ć 5!) / 5!
= 10 Ć 9 Ć 8 Ć 7 Ć 6
10! / 5! = 30240
Problem 8 :
21! / 14!
Solution :
21! / 14! = (21 Ć 20 Ć 19 Ć 18 Ć 17 Ć 16 Ć 15 Ć 14!) / 14!
= 21 Ć 20 Ć 19 Ć 18 Ć 17 Ć 16 Ć 15
21! / 14! = 586051200
Problem 9 :
9! / 3! 6!
Solution :
9! / 3! 6! = (9 Ć 8 Ć 7 Ć 6!) / 6! (3 Ć 2 Ć 1)
= (9 Ć 8 Ć 7) / (3 Ć 2 Ć 1)
= 504 / 6
9! / 3! 6! = 84
Problem 10 :
12! / 8! 4!
Solution :
12! / 8! 4! = (12 Ć 11 Ć 10 Ć 9 Ć 8!) / 8! (4 Ć 3 Ć 2 Ć1)
= (12 Ć 11 Ć 10 Ć 9) / (4 Ć 3 Ć 2 Ć1)
= 11880 / 24
12! / 8! 4! = 495
Problem 11 :
7! /2! 5! + 7! / 4! 3!
Solution :
7! /2! 5! + 7! / 4! 3! = (7 Ć 6 Ć 5!) / 5! (2 Ć 1) + (7 Ć 6 Ć 5 Ć 4!) / 4! (3 Ć 2 Ć 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 Ć 14 Ć 13 Ć 12 Ć 11 Ć 10 Ć 9!) / 9! (6 Ć 5 Ć 4 Ć 3 Ć 2 Ć1)
= (15 Ć 14 Ć 13 Ć 12 Ć 11 Ć 10) / (6 Ć 5 Ć 4 Ć 3 Ć 2 Ć1)
= 3603600 / 720
15! / 9! 6! = 5005
15! / 10! 5! = (15 Ć 14 Ć 13 Ć 12 Ć 11 Ć 10!) / 10! (5 Ć 4 Ć 3 Ć 2 Ć1)
= (15 Ć 14 Ć 13 Ć 12 Ć 11) / (5 Ć 4 Ć 3 Ć 2 Ć1)
= 360360 / 120
15! / 10! 5! = 3003
15! / 9! 6! + 15! / 10! 5! = 5005 + 3003
15! / 9! 6! + 15! / 10! 5! = 8008
Problem 13 :
2 Ć 5! / 2! 3!
Solution :
2 Ć 5! / 2! 3! = 2 Ć (5 Ć 4 Ć 3!) / 3! (2 Ć 1)
= 2 Ć (5 Ć 4) / 2
2 Ć 5! / 2! 3! = 20
Problem 14 :
3 Ć 11! / 7! 4!
Solution :
3 Ć 11! / 7! 4! = 3 Ć (11 Ć 10 Ć 9 Ć 8 Ć 7!) / 7! (4 Ć 3 Ć 2 Ć 1)
= 3 Ć (11 Ć 10 Ć 9 Ć 8) / (4 Ć 3 Ć 2 Ć 1)
= 3 Ć 330
3 Ć 11! / 7! 4! = 990
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