" Prove that,"'C_(+)+^()C_(n)=^(oo-1)C_(r)

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

Prove that : .^(n-1)C_(r)+.^(n-2)C_(r)+.^(n-3)C_(r)+.........+.^(r)C_(r)=.^(n)C_(r+1) .

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

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

If n is a positive integer then using the indentiy (1+x)^(n)=(1+x)^(3)(1+x)^(n-3) , prove that ""^(n)C_(r)=""^(n-2)C_(r)+3*""^(n-3)C_(r-1)*""^(n-3)C_(r-2)+""^(n-3)C_(r-3)

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

Prove that : (""^(n)C_(r+1))/(""^(n)C_(r))=(n-r)/(r+1)

Prove that : (""^(n)C_(r+1))/(""^(n)C_(r))=(n-r)/(r+1)