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

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

Statement-1: If underset(r=1)overset(n)(sum)r^(3)((.^(n)C_(r))/(.^(n)C_(r-1)))^(2)=196 , then the sum of the coeficients of powerr of xin the expansion of the polynomial (x-3x^(2)+x^(3))^(n) is -1. Statement-2: (.^(n)C_(r))/(.^(n)C_(r-1))=(n-r+1)/(r) AA n in N and r in W .

Prove that Sigma_(r=0)^(n) 3^(r)""^(n)C_(r)=4^(n)

Prove that Sigma_(r=0)^(n) ""^(n)C_(r).3^(r)=4^(n)

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

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

Prove that Sigma_(r=1) ""^(n)C_(r).3^(r)=4^(n)

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))

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

Prove that sum_(r=0)^(n) 3^( r" "n)C_(r ) =4^(n) .