Answer to Question #251146 in Discrete Mathematics for suhaa

Question

Prove that n•P(n−1,n−1)= P(n,n).

Answer 1

Solution:

$LHS=n.P(n-1,n-1) \\=n\times \dfrac{(n-1)!}{(n-1-n+1)!} \\=n\times \dfrac{(n-1)!}{(0)!} \\=n\times \dfrac{(n-1)!}{1} \\=n(n-1)! \\=n! \\=\dfrac{n!}{0!} \\=\dfrac{n!}{(n-n)!} \\=P(n,n) \\=RHS$

Hence, proved.