# FACTORIALS

Factorials :

The continued product of first n natural numbers is called the “n factorial” and is denoted by n!

i.e. n! = 1 × 2 × 3 × 4 × … × (n − 1) × n

5 ! = 1 × 2 × 3 × 4 × 5 = 120

Zero Factorials :

We will require zero factorial in the latter sections of this chapter. It does not make any sense to define it as the product of the integers from 1 to zero. So,

we define 0! = 1.

Let us see some example problems to understand the above concept.

Example 1 :

Simplify 5!/6!

Solution :

5! can be written as 5 ⋅  3  2  1

6! can be written as 6 ⋅ ⋅  3  2  1

5!/6!  =  (⋅  3  2  1) / (⋅ ⋅  3  2  1)

=  1/6

Example 2 :

Simplify (3!)²/9

Solution :

3! can be written as 3  2  1

(3!)²/9  =  ( 2  1)² / 9

=  9 ⋅ 4  1 / 9

=  4

Example 3 :

Simplify (3! ⋅ 4!/ 5!

Solution :

3! can be written as 3  2  1

4! can be written as 4 ⋅  2  1

5! can be written as 5 ⋅ 4 ⋅  2  1

(3! ⋅ 4!/ 5!  =  ( 2  1) (4 ⋅  2  1) / (⋅ 4 ⋅  2  1)

=  6/5

Example 4 :

Simplify 9! /(53!)

Solution :

9! can be written as 9  8  7  6  5!

3! can be written as  2  1

9! /(53!)  =  ( 8  7  6  5!) / (5! ⋅  2  1)

=  3 ⋅ 4 ⋅ 7 ⋅ 6

=  504

Example 5 :

Simplify (2 ⋅ 3)! /3!

Solution :

(2 ⋅ 3)! /3!  =  6! / 3!

6! can be written as  =  6  5  4  3  2  1

3! can be written as  =  3  2  1

6! / 3!  =  (6  5  4  3  2  1) / (3  2  1)

=  6  5  4

=  120

Example 6 :

Simplify 88! / 90!

Solution :

88! / 90!  =  88! / 90  89  88!

=  1/90  89

=  1/8010

Example 7 :

Simplify 77! 2! / 90!

Solution :

88! / 90!  =  88! / 90  89  88!

=  1/90  89

=  1/8010

Example 8 :

Simplify 93! / 89! 4!

Solution :

93! / 89! 4!  =  93  92  91  90  89! / 89!   4  3  2  1

=  31  23  91  45

=  2919735

Example 9 :

Simplify 21! / 14!

Solution :

21! / 14!  =  21  20  19  18  17  16  15  14!/14!

=  586051200

Example 10 :

Simplify 8! / 5!

Solution :

8! / 5!  =  8  7  6  5!/5!

=  336

After having gone through the stuff given above, we hope that the students would have understood "Factorials".

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