Prove that : `(C(2n,n))/(n+1)` is an integer

Using permutation or otherwise, prove that (n^2)!/(n!)^n is an integer, where n is a positive integer. (JEE-2004]

Using permutation or otherwise, prove that (n^2)!/(n!)^n is an integer, where n is a positive integer. (JEE-2004]

Using permutation or otherwise, prove that (n^2)!/(n!)^n is an integer, where n is a positive integer. (JEE-2004]

Prove that : If n is a positive integer then prove that i) C_(0)+C_(1)+C_(2)+……+C_(n)=2^(n)

Prove that : If n is a positive integer, then prove that C_(0)+(C_(1))/(2)+(C_(2))/(3)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1).

Prove that : If n is a positive integer and x is any nonzero real number, then prove that C_(0)+C_(1)(x)/(2)+C_(2).(x^(2))/(3)+C_(3).(x^(3))/(4)+….+C_(n).(x^(n))/(n+1)=((1+x)^(n+1)-1)/((n+1)x)

If n is an odd integer, prove that n^(2) is also an odd integer.

Prove that (1+i)^(2n)+(1-i)^(2n) = 0 if n is an odd integer and, equal to 2^(n+1)/((-1)^(n//2) if n is an even integer.

Prove that 2^n gt n for all positive integers n.

Prove that 2^n > n for all positive integers n.