Property :- (iii) `nC_r+nC_(r-1)=(n+1)C_r`

Property: ( i )nC_(r)=nC_(n-r)( ii) n(C_(r))/(r+1)=(n+1)(C_(r+1))/(n+-1)

Prove that "^nC_r + 2. ^nC_(r-1) + ^nC_(r-2) = ^(n+2)C_r

If n is a positive integer then ^nC_r+^nC_(r+1)=^(n+1)^C_(r+1) Also coefficient of x^r in the expansion of (1+x)^n=^nC_r. In an identity in x, coefficient of similar powers of x on the two sides re equal. On the basis of above information answer the following question: If n is a positive integer then ^nC_n+^(n+1)C_n+^(n+2)C_n+.....+^(n+k)C_n= (A) ^(n+k+1)C_(n+2) (B) ^(n+k+1)C_(n+1) (C) ^(n+k+1)C_k (D) ^(n+k+1)C_(n-2)

Property (iv) If ^nC_x=^nC_y ; then either x=yx=y or x+y=nx+y=n (v) r.nC_r=n. (n-1)C_(r-1)r.nC_r=n. (n-1)C_(r-1)

Write the expression ^nC_(r+1)+^(n)C_(r-1)+2xx^(n)C_(r) in the simplest form.

If f(n) = sum_(r=1)^n [r^2(nC_r-nC_(r-1))+(2r+1)nC_r], then f(50) is

If f(n) = sum_(r=1)^n [r^2(nC_r-nC_(r-1))+(2r+1)nC_r], then f(50) is

^nC_r+^nC_(r+1)+^nC_(r+2) is equal to (2lerlen) (A) 2^nC_(r+2) (B) 2^(n+1)C_(r+1) (C) 2^(n+2)C_(r+2) (D) none of these

det[[ The value of the determinant nC_(r-1),nC_(r),(r+1),n+2C_(r+1)nC_(r),nC_(r+1),(r+2),n+2C_(r+2)nC_(r+1),nC_(r+2),(r+3),n+2C_(r+3)]] is

If n is a positive integer such that (1+x)^n=^nC_0+^nC_1+^nC_2x^2+…….+^nC_nx^n, for epsilonR . Also ^nC_r=C_r On the basis o the above information answer the following questions the value of ^mC_r.^nC_0+^mC_(r-1).^nC_1+^mC_(r-2).^nC_2+….+^mC_1.^nC_(r-1)+^nC_0^nC_r where m,n, r are positive interges and rltm,rltn= (A) ^(mn)C_r (B) ^(m+n)C_r (C) 0 (D) 1