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`

Prove that 2^(k)(n,0)(n,k)-2^(k-1)(n,1)(n-1,k-1)+2^(k-2)(n,2)(n-2,k-2)-....+(-1)^(k)(n,k)(n-k,0)=(n,k)

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.

If n is a positive integer,then nC_(1)+nC_(2)+...+nC_(n)=2^(n)-1

nC_(0)-(1)/(2)(^(^^)nC_(1))+(1)/(3)(^(^^)nC_(2))-....+(- 1)^(n)(nC_(n))/(n+1)=

(n+2)nC_0(2^(n+1))-(n+1)nC_1(2^(n))+(n)nC_2(2^(n-1))-.... is equal to

Show that ""^(n)C_(0) +""^((n+1))C_(1) +""^((n+2))C_(2) +…+""^((n+k))C_(k) =""^((n+k+1))C_(k)

Suppose m and n are positive integers and let S=sum_(k=0)^(n)(-1)^(k)(1)/(k+m+1)(nC_(k)) and T=sum_(k=0)^(m)(-1)^(k)1(k+n+1)(mC_(k)) then S-T is equal to

If n is a positive integer SC_(k)=^(n)C_(k), find the value of (sum_(k=1)^(n)(k^(3))/(n(n+1)^(2)*(n+2))((C_(k))/(C_(k)-1))^(2))^(-1)