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

Prove that sum_(r=0)^n^n C_rsinr xcos(n-r)x=2^(n-1)sin(n x)dot

Prove that sum_(r=0)^n(-1)^r^n C_r[1/(2^r)+3/(2^(2r))+7/(2^(3r))+(15)/(2^(4r))+ u ptomt e r m s]=(2^(m n)-1)/(2^(m n)(2^n-1))

Prove that sum_(r=0)^n 3^r n Cundersetr = 4^n .

Find the sum sum_(r=1)^n r^2(^n C_r)/(^n C_(r-1)) .

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

Prove that sum_(r=0)^(2n)r(.^(2n)C_r)^2=n^(4n)C_(2n) .

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

Prove that sum_(k=0)^(n) (-1)^(k).""^(3n)C_(k) = (-1)^(n). ""^(3n-1)C_(n)

If x+y=1, prove that sum_(r=0)^n r* ^nC_r x^r y^(n-r)=nxdot

Prove that sum_(r=1)^n(-1)^(r-1)(1+1/2+1/3++1/r)^n C_r=1/n .