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

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))

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

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

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

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

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_r(-1)^r[i^r+i^(2r)+i^(3r)+i^(4r)]=2^n+2^(n/2+1)cos(npi//4),w h e r ei=sqrt(-1)dot

Probability if n heads in 2n tosses of a fair coin can be given by prod_(r=1)^n((2r-1)/(2r)) b. prod_(r=1)^n((n+r)/(2r)) c. sum_(r=0)^n((^n C_r)/(2^n)) d. (sumr=0n(^n C_r)^2)/(sumr=0 2n(^(2n)C_r)^)

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