Prove, by induction, that `(nC_0)/x -(nC_1)/(x+1)+(nC_2)/(x+2)-...........+(-1)^n * (nC_n)/(x+n)=(n!)/(x(x+1)(x+2)......(x+n)),x in R ,x leq theta` for all `n in N`.

(nC_(0))^(2)-(nC_(1))^(2)+(nC_(2))^(2)+....+(-1)^(n)(nC_(n))^(2)

Prove that (nC_(0))/(x)-(n_(C_(0)))/(x+1)+(^nC_(1))/(x+2)-...+(-1)^(n)(n_(n))/(x+n)=(n!)/(x(x+1)...(x-n)) where n is any positive integer and x is not a negative integer.

Show that : ""^nC_0*m-""^nC_1*(m-1)+""^nC_2*(m-2)-...+(-1)^n*""^nC_n*(m-n)=0

nC_(0)-(1)/(2)(^(^^)nC_(1))+(1)/(3)(^(^^)nC_(2))-....+(- 1)^(n)(nC_(n))/(n+1)=

The value of ("^n C_0)/n + ("^nC_1)/(n+1) + ("^nC_2)/(n+2) +....+ ("^nC_ n)/(2n) is equal to

The value of (nC_(0))/(n)+(nC_(1))/(n+1)+(nC_(2))/(n+2)+......+(nC_(n))/(2n) is equal to

Prove that 2*""^nC_0+2^2*(""^nC_1)/2+2^3*(""^nC_2)/3+...2^(n+1)*(""^nC_n)/(n+1)=(3^(n+1)-1)/(n+1)

Prove that: (1)/(2)nC_(1)-(2)/(3)nC_(2)+(3)/(4)nC_(3)-(4)/(5)nC_(4) +....+((-1)^(n+1)n)/(n+1)*nC_(n)=(1)/(n+1)

Prove that ""^nC_0+3""^nC_1+5""^nC_2+........+(2n+1)""^nC_n= (n+1)2^n .

Find the value of (1+x)^(n)+nc_(1)(1+x)^(n-1)*(1-x)+^(n)C_(2)(1+x)^(n-2)(1-x)^(2)+.........+(1-x)^(n)