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

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

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

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

Prove that nC_(r)+n-1C_(r)+n-2C_(r)+.......+rC_(r)=n+1C_(1)

"^mC_1=^nC_2 , then -

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

nCr + 2.nC (r-1) + nC (r-2) = (n + 2) Cr (2 <= r <= n)

Prove that ^nC_(r)+^(n-1)C_(r)+...+^(r)C_(r)=^(n+1)C_(r+1)