Prove that `n! (n+2) = n! +(n+1)!`.

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

Prove that 2^(n)>n,n in N

Prove that [n+1/2]^(n)>(n!)

Prove that ((n + 1)/(2))^(n) gt n!

For a fixed positive integer n prove that: D = |[n!, (n+1)!, (n+2)!],[(n+1)!,(n+2)!,(n+3)!],[(n+2)!,(n+3)!,(n+4)!]|= ( n !)^3(2n^3+8n^2+10n+4 )

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

Prove that ((2n+1)!)/(n!)=2^(n)[1.3.5.....(2n-1)*(2n+1)]

Prove that ((2n+1)!)/(n!)=2^(n){1.3.5(2n-1)(2n+1)}