If `x+y=1,` prove that `sum_(r=0)^n^n C_r x^r y^(n-r)dot`

If x+y=1, then sum_(r-0)^(n)rnC_(r)x^(r)y^(n-r) equals

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

If ""^(n)C_(0), ""^(n)C_(1),..., ""^(n)C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) and p + q = 1 , then sum_(r=0)^(n) ""r.^(n)C_(r) p^(r) q^(n-r) =

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))

If x+y= 1, then value of sum_(r=0)^(n)(r)(""^(n)C_(r))x^(n-r)y^(r) is

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