If `(1+x)^(n) = overset(n)underset(r=0)Sigma C_(r)x^(r )`, then prove that `C_(1)+2C_(2)+3C_(3)+…..+nC_(n)=n2^(n-1)`.

If (1+x)^(n) = overset(n)underset(r=0) Sigma C_(r) x^(r ) , show that C_(0)+(C_(1))/(2)+….+(C_(n))/(n+1)=(2^(n+1)-1)/(n+1)

Prove that nC_r = nC_(n-r)

Prove that C_0C_2+C_1C_3+…+C_(n-2)C_n=^(2n)C_(n-2)

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

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

Prove that C_0+C_2/3+C_4/5+...=(2^n)/(n+1)