Factorial Notation

For any natural number from 1 to n that is n factorial or n! Is expressed as,

1×2×3×....×(n-2)×(n-1)×n.

Note: the factorial notation that that can also be defined as a product of natural numbers from n to 1.

i.e, n!=n×(n-1)×(n-2)×...×3×2×1

5!=5×4×3×2×1=120

*Properties of factorial notation:-

1) 0! = 1

2) (m×n)! ≠ m!×n!

3) (m+n)! ≠ m!+n!

4) (m-n)! ≠ m!-n!

5) (m÷n)! ≠ m!÷n!

Ex 1) Find the value of 6!

Answer-

6 x 5 x 4 x 3 x 2 x 1= 720

Ex 2) Find n if (n+6)!=56(n+4)!

Solution-

Factorial Notation 11th Class Maharashtra Board EXERCISE 3.2

Find n if (n+6)!=56(n+4)!

Ex Show that $\frac{12!}{5! 7!} + \frac{12!}{6! 6!} = \frac{13!}{6! 7!}$ $\frac{\text {1}2!}{\text{5}!7!}+\frac{\text{1}2!}{\text{6}!6!}=\frac{\text{1}3!}{\text{6}!7!}$

Solution:

L.H.S. =

\frac{12!}{5! 7!} + \frac{12!}{6! 6!}

$\frac{\text{1}2!}{\text{5}!7!}+\frac{\text{1}2!}{\text{6}!6!}$

12! [\frac{1}{5! 76!} + \frac{1}{5! 66!}]

$12!\left[\frac{\text{1}}{\text{5}!76!}+\frac{\text{1}}{\text{5}!66!}\right]$

(since, 7!=7×6! 6!=6×5!)

\frac{12!}{5! 6!} [\frac{13}{6 7}]

$\frac{\text{12!}}{\text{5!}6!}\left[\frac{\text{13}}{\text{6 7}}\right]$

\frac{12! 13}{(5! 6)*(6! 7)}

$\frac{\text{12! 13}}{\text{(5! 6)*(6! 7)}}$

\frac{13!}{6! 7!}

$\frac{\text{13!}}{\text{6! 7!}}$

=R.H.S

Factorial Notation