Prove that `overset(n) underset(r=0)Sigma^(n)C_(r) sin rx cos (n-r)x=2^(n-1) sin (nx)`.

Find the sum overset(n)underset(r=1) Sigma (r""^(n)C_(r))/(""^(n)C_(r-1))

Prove that underset(r=0)oversetnsum3^r^nC_r=4^n

Find the coefficient of x^(25) in expansion of expression overset(50)underset(r=0)Sigma ""^(50)C_(r)(2x-3)^(r )(2-x)^(50-r)

Prove that nC_r = nC_(n-r)