If `(1+x)^n=sum_(r=0)^n^n C_r` , show that `C_0+(C_1)/2++(C_n)/(n+1)=(2^(n+1)-1)/(n+1)` .

If (1+2x+x^2)^n=sum_(r=0)^(2n)a_r x^r ,then a= (^n C_2)^2 b. ^n C_rdot^n C_(r+1) c. ^2n C_r d. ^2n C_(r+1)

If C_(r) = .^(n)C_(r) then prove that (C_(0) + C_(1)) (C_(1) + C_(2)) "….." (C_(n-1) + C_(n)) = (C_(1)C_(2)"…."C_(n-1)C_(n))(n+1)^(n)//n!

If sum_(r=0)^(2n)a_r(x-2)^r=sum_(r=0)^(2n)b_r(x-3)^ra n da_k=1 for all kgeqn , then show that b_n=^(2n+1)C_(n+1) .

Prove that ""^(n)C_r + ""^(n)C_(r-1) = ""^(n+1)C_r

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

If (1+x)^n=C_0+C_1x+C_2x^2++C_n x^n ,t h e nC_0C_2+C_1C_3+C_2C_4++C_(n-2)C_n= ((2n)!)/((n !)^2) b. ((2n)!)/((n-1)!(n+1)!) c. ((2n)!)/((n-2)!(n+2)!) d. none of these

.^((n-1)C_(r) + ^((n-1)) C_((r-1)) is

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

The value of sum_(r=1)^n(-1)^(r+1)("^n C r)/(r+1) is equal to a. -1/(n+1) b. 1/n c. 1/(n+1) d. n/(n+1)

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