Prove: `n C r ​ + n C r−1` ​ = `n+1 C r` ​ .

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

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

Prove that coefficient of x^(r ) in (1-x)^(-n) " is " ""^(n+r-1)C_(r )

Prove that: (i) r.^(n)C_(r) =(n-r+1).^(n)C_(r-1) (ii) n.^(n-1)C_(r-1) = (n-r+1) .^(n)C_(r-1) (iii) .^(n)C_(r)+ 2.^(n)C_(r-1) +^(n)C_(r-2) =^(n+2)C_(r) (iv) .^(4n)C_(2n): .^(2n)C_(n) = (1.3.5...(4n-1))/({1.3.5..(2n-1)}^(2))

Prove that sum_(r = 0)^n r^2 . C_r = n (n +1).2^(n-2)

Prove that sum_(r = 1)^n r^3 ((n_C_r)/(C_(r - 1)))^2 = (n (n + 1)^2 (n+2))/(12)

If (1+2x+x^2)^n=sum_(r=0)^(2n)a_r x^r ,then a_r is a. (.^nC_2)^2 b. .^n C_r .^n C_(r+1) c. .^(2n) C_r d. .^(2n) C_(r+1)

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

Prove that .^(n)C_(0) - .^(n)C_(1) + .^(n)C_(2) - .^(n)C_(3) + "……" + (-1)^(r) + .^(n)C_(r) + "……" = (-1)^(r ) xx .^(n-1)C_(r ) .

Prove that combinatorial argument that ^n+1C_r=^n C_r+^n C_(r-1)dot