`mC_r+mC_(r-1)nC_1+mC_(r-2)nC_2+...........+nC_r=(m+n)C_r`

nC_r+2. nC_(r-1)+nC_(r-2)=

Statement 1: ^m C_r+^m C_(r-1)(^nC_1)+^mC_(r-2)(^n C_2)+....+^n C_r=0 , if m+n lt r Statement 2: ^n C_r=0 , if n lt r

Statement 1:^mC_(r)+mC_(r-1)^(n)C_(1)+mC_(r-2)^(n)C_(2)+...+^(n)C_(r)=0, if m+n

Statement 1: ""^m C_r+ ""^m C_(r-1)(""^nC_1)+ ""^mC_(r-2)(""^n C_2)+....+ ""^n C_r=0 , if m+n lt r Statement 2: ""^n C_r=0 , if n lt r (a) Statement 1 and Statement 2, both are correct. Statement 2 is the correct explanation for Statement 1. (b) Statement 1 and Statement 2, both are correct. Statement 2 is not the correct explanation for Statement 1. (c) Statement 1 is true but Statement 2 is false. (d) Statement 2 is true but Statement 1 is false.

Statement-1 : The sum of the series ^nC_0. ^mC_r+^nC_1.^mC_(r-1)+^nC_2.^mC_(r-2)+......+^nC_r.^mC_0 is equal to ^(n+m)C_r, where C's and C's denotes the combinatorial coefficients in the expansion of (1 + x)^n and (1 + x)^m respectively, Statement-2: Number of ways in which r children can be selected out of (n + m) children consisting of n boys and m girls if each selection may consist of any number of boys and girls is equal to ^(n+m)C_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 of 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 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

Using Mathematical Induction, prove that : "^mC_0 ^nC_k+ ^mC_1 ^nC_(k-1)+..........+ ^mC_k ^nC_0= ^(m+n)C_k , where m, n, r are positive integers and "^pC_q = 0 for p < q.

Prove that ""^nC_r+4^nC_(r-1)+6^(n)C_(r-2)+4^nC_(r-3)+^nC_(r-4)=^(n+4)C_r