`C_0-C_1+C_2-C_3+......+(-1)^rC_r=((-1)^r(n-1)!)/(r!.(n-r-1)!)`

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

Prove that C_0C_r+C_1 C_(r+1)+ C_2 C_(r+2)+...............+c_(n-r) C_n=((2n)!)/((n-r)!(n+r)!)

Prove that C_0C_r+C_1 C_(r+1)+ C_2 C_(r+2)+...............+c_(n-r) C_n=((2n)!)/((n-r)!(n+r)!)

Show that: C_0C_r+C_1C_(r+1)+C_2C_(r+2)+….+C_(n-r)C_n= ((2n)!)/((n-r)!(n+r)!)

Prove that (r+1)^n C_r-r^n C_r+(r-1)^n C_2-^n C_3++(-1)^r^n C_r = (-1)^r^(n-2)C_rdot

If C_0, C_1, C_2 ,……..C_n are the coefficient in the expansion of (1 + x)^n then show that C_0 C_r + C_1 C_(r + 1) + C_2 C_(r + 2) + ………..+ C_(n-r).C_n = ((2n)!)/((n-r)!(n+r)!)

Given that C_r/(C_(r-1))=(n-r+1)/r Prove that (1+C_1/C_0)(1+C_2/C_1)...(1+C_n/(C_(n-1)))=(n+1)^n/(n!)

The roots of the equation |^x C_r^(n-1)C_r^(n-1)C_(r-1)^(x+1)C_r^n C_r^n C_(r-1)^(x+2)C_r^(n+1)C_r^(n+1)C_(r-1)|=0 are a) x=n b) x=n+1 c) x=n-1 d) x=n-2

The roots of the equation |^x C_r^(n-1)C_r^(n-1)C_(r-1)^(x+1)C_r^n C_r^n C_(r-1)^(x+2)C_r^(n+1)C_r^(n+1)C_(r-1)|=0 are a) x=n b) x=n+1 c) x=n-1 d) x=n-2

The roots of the equation |^x C_r^(n-1)C_r^(n-1)C_(r-1)^(x+1)C_r^n C_r^n C_(r-1)^(x+2)C_r^(n+1)C_r^(n+1)C_(r-1)|=0 are x=n b. x=n+1 c. x=n-1 d. x=n-2