If `n` and `k` are positive integers, show that `2^k(nC 0)(n k)-2^(k-1)(nC1)(n-1Ck-1)+2^(k-2)(nC2)((n-2k-2))_dot-...+ (-1)^k(nCk)+(n-kC0)=(nC k)w h e r e(n C k)` stands for `^n C_kdot`

If n and k are positive integers, show that 2^k( .^n C_0)(.^n C_k)-2^(k-1)(.^n C_1)(.^(n-1) C_k-1)+2^(k-2)(.^n C_2)((n-2k-2))_dot-...+ (-1)^k(^n C_k)+(.^(n-k) C_0)=(.^n C_k)w h e r e(.^n C_k) stands for .^n C_k.

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

Show that: .^nC_(n-r)+3.^nC_(n-r+1)+3.^nC_(n-r+2)+^nC_(n-r+3)=^(n+3)C_r .

Show that lim_(nrarr infty) sum_(k=0)^(n)(""^(n)Ck)/(n^k(k+3))=e-2

If k a n d n are positive integers and s_k=1^k+2^k+3^k++n^k , then prove that sum_(r=1)^m^(m+1)C_r s_r=(n+1)^(m+1)-(n+1)dot

If n is a positive integer then (1+x)^n=^nC_0 x^0+^nC_1 x^1+^nC_2^2+………+^nC_rx^r= sum _(r=0)^n ^nC_rx^r and (1-x)^n= ^nC_0x^0-^nC_1 x^1 +^nC_2-^nC_3x^3+………+(-1)^n ^nC_nx^n=sum_(r=0)^n (-1)^r ^nC_r x^r On the basis of above information answer the following question: If n is a positive integer then 1/((49)^n) - 8/((49)^n)(^(2n)C_1)+8^2/((49)^n)( ^(2n)C_2)- 8^3/((49)^n)(^(2n)c_3)+......+8^(2n)/((49)^n)= (A) -1 (B) 1 (C) (64/49)^n (D) none of these

If n is a positive integer and C_(k)=""^(n)C_(k) , then the value of sum_(k=1)^(n)k^(3)((C_(k))/(C_(k-1)))^(2) is :

If (nC_0)/(2^n)+2.(nC_1)/2^n+3.(nC_2)/2^n+....(n+1)(nC_n)/2^n=16 then the value of 'n' is

Find the value lim_(n rarr oo) sum_(k=2)^(n) cos^(-1) ((1 + sqrt((k -1) k(k + 1) (k + 2)))/(k(k + 1)))

If the value of (n + 2) . ""^(n)C_(0) *2^(n+1) - (n+1) * ""^(n)C_(1)*2^(n) + n* ""^(n)C_(2) * 2^(n-1) -... is equal to k(n +1) , the value of k is .