Show that `sum_(k=m)^n ^kC_r=^(n+1)C_(r+1)-^mC_(r+1)`

Statement-1 sum_(r=0)^(n) r ""^(n)C_(r) x^(r) (-1)^(r) = nx (1 - x)^(n -1) Statement-2: sum_(r=0)^(n)r ""^(n)C_(r) x^(r) (-1)^(r) =0

Deduce that: sum_(r=0)^(n)*^(n)C_(r)(-1)^(n)(1)/((r+1)(r+2))=(1)/(n+2)

Find the sum of sum_(r=1)^(n)(r^(n)C_(r))/(^nC_(r-1))

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))

Show that the HM of (2n+1)C_(-) and (2n+1)C_(-)(r+1) is (2n+1)/(n+1) xx of (2n)C_(r) Also show that sum_(r=1)^(2n-1)(-1)^(r-1)*(r)/(2nC_(r))=(n)/(n+1)

If k and n are positive integers and s_(k)=1^(k)+2^(k)+3^(k)+...+n^(k), then prove that sum_(r=1)^(m)m+1C_(r)s_(r)=(n+1)^(m+1)-(n+1)