Question

Permutation, prove it :
nP*n=2(nP*n-2)

Answer 1

To prove:-

$^n\!P_n=2\left(\!^n\!P_{n-2}\right)$

Solution:-

$\begin{aligned}&\sf{LHS}\\\\=\ &\ ^n\!P_n\\\\=\ &\dfrac {n!}{(n-n)!}\\\\=\ &\dfrac {2!\times n!}{2!\times 0!}\\\\=\ &2\left (\dfrac {n!}{2!}\right)\\\\=\ &2\left (\dfrac {n!}{(n-n+2)!}\right)\\\\=\ &2\left (\dfrac {n!}{(n-(n-2))!}\right)\\\\=\ &2\left (\!^n\!P_{n-2}\right)\\\\=\ &\sf{RHS}\end {aligned}$

Hence Proved!