Show ` . ^nC_r =^nC_(n-r) ` and also give an example demonstrating this.

If .^nP_r=^nP_(r+1) ,^nC_r=^nC_(r-1) then find value of r

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

If nC_(r)=nC_(r-1) and nP_(r)=nP_(r+1), then the value of n 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 the series sum_(r=1)^n r^2.C_r= (A) 1 (B) (-1)^(n/2).n!/(n/2!)^2 (C) (n-1).^(2n)C_n+2(2n) (D) n(n+1)2^(n-2)

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 for any aepsilon R the value of the expression a-(a-1)C_1+(a-2)C-2-(a-3)C_3+.+(1)^n(a-n)C_n= (A) 0 (B) a^n.(-1)^n.^(2n)C_n (C) [2a-n(n+1)[.^(2n)C_n (D) none of these