Prove that(i) P(n,n) = 2P(n,n-2)(ii) P(n,r) = (n-r+1)P(n,r-1) - Maths - Permutations and Combinations

Question

Prove that(i) P(n,n) = 2P(n,n-2)(ii) P(n,r) =
(n-r+1)P(n,r-1)

Ajanta Trivedi · Accepted Answer

since ${}^{n}{P_{r}} = \frac{n!}{(n - r)!}$ .................(1)

(i)

$P (n, n) = {}^{n}{P_{n}} = \frac{n!}{(n - n)!} = \frac{n!}{0!} = \frac{n!}{1} = 2 * \frac{n!}{2!} [∵ 2! = 2]$

$= 2 * \frac{n!}{(n - {n - 2})!} = 2 * {}^{n}{P_{n - 2}} = 2 * P (n, n - 2)$

2.

$P (n, r) = {}^{n}{P_{r}} = \frac{n!}{(n - r)!} = (n - r + 1) * \frac{n!}{(n - r + 1) (n - r)!} = (n - r + 1) * \frac{n!}{(n - r + 1)!} = (n - r + 1) * \frac{n!}{(n - {r - 1})!} = (n - r + 1) * {}^{n}{P_{r - 1}}$

$= (n - r + 1) * P (n, r - 1)$

= RHS

hope this helps you.

cheers!!