Factorial Notation and Deduction - Permutations and Combination

Factorial Notation

0! = 1

1! = 1

2! = 1 x 2

3! = 1 x 2 x 3

4! = 1 x 2 x 3 x 4

n! = 1 x 2 x 3 x 4 x .......... x n

Defintion: The continued product of first n natural number is called the "n factorial" and denoted by n!. Also n! is defined for positive integer only.

For Example:
Factorial of 4 = 4! = 1 x 2 x 3 x 4 = 24
Factorial of 5 = 5! = 1 x 2 x 3 x 4 x 5 = 120

Deduction in Factorial Notation

Recursive Defintion: Factorials can be expressed recursively by recognizing that n! includes (n-1)! as a sub product. This recursive property hepls simplify and understand factorial expressions.

n! = n x (n-1) x (n-2) x (n-3) x ...... x 3 x 2 x 1
n! = n x [(n-1) x (n-2) x (n-3) x ...... x 3 x 2 x 1]
n! = n x (n-1)!

Thus, 5! = 5 x 4!

Practice Question

Question 1. if $\frac{1}{6!} \text{+} \frac{1}{7!} \text{=} \frac{x}{8!}\text{;}$ find x.

Solution:
We have,
$\frac{1}{6!} \text{ + } \frac{1}{7!} \text{=} \frac{x}{8!}$

$\text{ or, } \frac{8 * 7}{8 * 7 * 6!} \text{ + } \frac{8}{8 * 7!} \text{ = } \frac{x}{8!}$

$\text{ or, }\frac{56}{8!} \text{ + } \frac{8}{8!} \text{ = } \frac{x}{8!}$
or, 56 + 8 = x
or, x = 64