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`

.^(n)C_(r)+2.^(n)C_(r-1)+.^(n)C_(r-2)=

Find the sum of sum_(r=1)^n(r^n C_r)/(^n C_(r-1) .

""^(n)C_(r+1)+^(n)C_(r-1)+2.""^(n)C_(r)=

For ""^(n) C_(r) + 2 ""^(n) C_(r-1) + ""^(n) C_(r-2) =

Show that sum_(k=m)^n ^kC_r=^(n+1)C_(r+1)-^mC_(r+1)

Statement-1: sum_(r =0)^(n) (r +1)""^(n)C_(r) = (n +2) 2^(n-1) Statement -2: sum_(r =0)^(n) (r+1) ""^(n)C_(r) x^(r) = (1 + x)^(n) + nx (1 + x)^(n-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 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

Show that: .^nC_(n-r)+3.^nC_(n-r+1)+3.^nC_(n-r+2)+^nC_(n-r+3)=^(n+3)C_r .

""^(n) C_(r+1)+2""^(n)C_(r) +""^(n)C_(r-1)=

Statement -1: sum_(r=0)^(n) r(""^(n)C_(r))^(2) = n (""^(2n -1)C_(n-1)) Statement-2: sum_(r=0)^(n) (""^(n)C_(r))^(2)= ""^(2n)C_(n)