Prove that ​ (i) n!/r! = n(n – 1) (n – 2) …. (r + 1) (ii) (n - r + 1) n!/(n-r+1)! = n!/(n-r)!

Question

Prove that

(i) \(\frac{n!}{r!}\) = n(n – 1) (n – 2) …. (r + 1)

(ii) (n - r + 1) \(\frac{n!}{(n-r+1)!}=\frac{n!}{(n-r)!}\) 

(iii)  \(\frac{n!}{r!(n-r)!}\) + \(\frac{n!}{(r-1)!(n-r+1)!}\) = \(\frac{(n+1)!}{r!(n-r+1)!}\)

Answer 1

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

Formula : n! = n x (n-1)!

L.H.S. = \(\frac{n!}{r!}\) 

Writing (n!) in terms of (r!) by using above formula,

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

Cancelling (r!), 

= n(n - 1)(n - 2)…. (r + 1) 

= R.H.S. 

∴ LHS = RHS 

Note : In permutation and combination r is always less than n, so we can write n! in terms of r! by using given formula.

= R.H.S. 

∴LHS = RHS