Prove that `(n !)^2` < `n^n n!` < `(2n)!`, for all positive integers n.

Prove that (n !+1) is not divisible by any natural number between 2 and n

Prove that C_(0)^(2) + C_(1)^(2) + C_(2)^(2) +…+C_(n)^(2) = (2n!)/(n!)^(2)

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

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

Prove that 4n^(2)-1 is an odd number for all n in N

Prove that ((2n)!)/(n!) =2^(n) (1,3,5,……..(2n-1)) .

Prove that 3^(2n)-1 is divisible by 8.

For any positive integer (m,n) (with ngeqm ), Let ((n),(m)) =.^nC_m Prove that ((n),(m)) + 2((n-1),(m))+3((n-2),(m))+....+(n-m+1)((m),(m))

Prove that: ((2n)!)/(n !)={1. 3. 5 (2n-1)}2^ndot