|(1)/((2)(n!)/((n-r)!))=n(n-1)(n-2)...(n-(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) (n!)/(r!)=n(n-1)(n-2)...(r+1) (ii) (n-r+1)*(n!)/((n-r+1)!)=(n!)/((n-r)!) (iii) (n!)/(r!(n-r)!)+(n!)/((r-1)!(n-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

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

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

Prove that: (i) (n!)/(r!) = n(n-1) (n-2)......(r+1) (ii) (n-r+1). (n!)/((n-r+1)!) = (n!)/((n-r)!)

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

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

If Delta_(r )=|(r,2r-1, 3r-2),((n)/(2), n-1, a),((1)/(2)n(n-1), (n-1)^(2), (1)/(2)(n-1)(3n-4))| , then the value of sum_(r=1)^(n-1)Delta_(r )