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Examples

(1) Evaluate : $\frac{20!}{17!}$

Soln. We have, $\frac{20!}{17!}$ = $\frac{20 \times 19 \times 18 \times 17!}{17!}$ = 20 $\times$ 19 $\times$ 18 = 6840.

(2) If repetitions are not allowed, find how many four digit number using even digits can be formed.

Soln. Even digits = 0, 2, 4, 6, 8.

Thousandth Place	Hundredth Place	Tenth Place	Unit Place
Can Use : 2, 4, 6, 8	Can use : remaining 4 nos. ( Including zero )	Can use : remaining 3 nos.	Can use : remaining 2 nos.
4 ways	4 ways	3 ways	2 ways

$\therefore$ Total required nos. = 4 $\times$ 4 $\times$ 3 $\times$ 2 = 96.

(3) What is the value of ‘n’, if $_{n}C_{5}$ = 30 ( $_{n}C_{4}$ ) ?

Soln. $_{n}C_{5}$ = 30 ( $_{n}C_{4}$ )

$\therefore$ $\frac{n(n-1)(n-2)(n-3)(n-4)}{5!}$ = 30 $\times$ $\frac{n(n-1)(n-2)(n-3)}{4!}$

$\therefore$ $\frac{n(n-1)(n-2)(n-3)(n-4)}{n(n-1)(n-2)(n-3)}$ = 30 $\times$ $\frac{5!}{4!}$

$\therefore$ n – 4 = 150

$\therefore$ n = 154