Prove that `sum_(r=0)^ssum_(s=1)^n^n C_s^n C_r=3^n-1.`

Prove that sum_(r=0)^(n)C_(r)sin rx cos(n-r)x=2^(n-1)sin(nx)

The value of sum_(r=0)^(n)sum_(s=1)^(n)*^(n)C_(5)*^(s)C_(r) is

Prove that sum_(n)^(r=0) ""^(n)C_(r)*3^(r)=4^(n).

If (1+x)^n=sum_(r=0)^n C_rx^r then prove that sum_(r=0)^n (C_r)/((r+1)2^(r+1))=(3^(n+1)-2^(n+1))/((n+1)2^(n+1))

Prove that sum_(r=0)^(n)r(n-r)C_(r)^(2)=n^(2)(^(2n-2)C_(n))

Prove that (3!)/(2(n+3))=sum_(r=0)^(n)(-1)^(r)((^nC_(r))/(r+3C_(r)))

If C_(0), C_(1), C_(2),…, C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) , then sum_(r=0)^(n)sum_(s=0)^(n)(C_(r) +C_(s))