1 PROVE n! /(n-r)!=n(n-1)(n-2) (n-r+1) 2 (n-r+1) n!(n-r+1)!=n!/(n-r)! - Maths - Permutations and Combinations

Question

1 PROVE n! /(n-r)!=n(n-1)(n-2) (n-r+1)
2 (n-r+1) n!(n-r+1)!=n!/(n-r)!

Manmeet Singh · Accepted Answer

Due to paucity of time, we are providing solution to only one of your query. Try the other yourself, if you face problem, then do get back to us.

To prove : $\frac{n!}{(n - r)!} = n (n - 1) (n - 2) ... (n - r + 1)$ ...(1)

R.H.S. = $n (n - 1) (n - 2) ... (n - r + 1)$

On multiplying by $\frac{(n - r)!}{(n - r)!}$ , we get

⇒ R.H.S.

$= {n (n - 1) (n - 2) ... (n - r + 1)} \frac{(n - r)!}{(n - r)!} = {n (n - 1) (n - 2) ... (n - r + 1)} \frac{(n - r) (n - r - 1) (n - r - 2) ...3.2.1}{(n - r)!} = \frac{{n (n - 1) (n - 2) ... (n - r + 1) (n - r) (n - r - 1) (n - r - 2) ...3.2.1}}{(n - r)!} = \frac{{n (n - 1) (n - 2) ...3.2.1}}{(n - r)!} = \frac{n!}{(n - r)!}$

1.PROVE n! /(n-r)!=n(n-1)(n-2)...(n-r+1) 2. (n-r+1).n!(n-r+1)!=n!/(n-r)!

1.PROVE n! /(n-r)!=n(n-1)(n-2)...(n-r+1)

2. (n-r+1).n!(n-r+1)!=n!/(n-r)!