" (i) "(n!)/(r!)=n(n-1)(n-2)...(r+1)

Prove that (n!)/(r!)=n(n-1)(n-2)dots(r+1)

If (1-x)^(-n)=a_(0)+a_(1)x+a_(2)x^(2)+...+a_(r)x^(r)+..., then a_(0)+a_(1)+a_(2)+...+a_(r) is equal to (n(n+1)(n+2)...(n+r))/(r!)((n+1)(n+2)...(n+r))/(r!)(n(n+1)(n+2)...(n+r-1))/(r!) none of these

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))

If (1-x)^(-n)=a_0+a_1x+a_2x^2+...+a_r x^r+ ,t h e na_0+a_1+a_2+...+a_r is equal to (n(n+1)(n+2)(n+r))/(r !) ((n+1)(n+2)(n+r))/(r !) (n(n+1)(n+2)(n+r-1))/(r !) none of these

(ii) (n!)/((n-r)!r!)+(n!)/((n-r+1)!(r-1)!)=((n+1)!)/(r!(n-r+1)!)

If (1-x)^(-n)=a_0+a_1x+a_2x^2++a_r x^r+ ,t h e na_0+a_1+a_2++a_r is equal to a. (n(n+1)(n+2)(n+r))/(r !) b. ((n+1)(n+2)(n+r))/(r !) c. (n(n+1)(n+2)(n+r-1))/(r !) d. none of these

Prove that (n!)/(r!(n-r)!)+(n!)/((r-1)!(n-r+1)!) =((n+1)!)/(r!(n-r+1)!)

Prove that (n!)/(r!(n-r)!)+(n!)/((r-1)!(n-r+1)!)= ((n+1)!)/ (r!(n-r+1)!) .

Prove that (n!)/(r!(n-r)!)+(n!)/((r-1)!(n-r+1)!) =((n+1)!)/(r!(n-r+1)!)

If 1<=r<=n, then n^(n-1)C_(r)=(n-r+1)^(n)C_(r-1)