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)!)`

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)!)

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

Find the coefficient of x^(n-r) in the expansion of (x+1)^n (1+x)^n . Deduce that C_0C_r+C_1C_(r-1)+......+C_(n-r) C_n= ((2n!))/((n+r)!(n-r)!) .

Prove that : For n = 0, 1, 2, 3, ………., n, prove that C_(0).C_(r)+C_(1).C_(r+1)+C_(2).C_(r+2)+….+C_(n-r).C_(n) =""^(2n)C_((n+r)) and hence deduce that C_(0).C_(1)+C_(1).C_(2)+C_(2).C_(3)+……..+C_(n-1).C_(n)=""^(2n)C_(n+1)

Prove that : For n = 0, 1, 2, 3, ………., n, prove that C_(0).C_(r)+C_(1).C_(r+1)+C_(2).C_(r+2)+….+C_(n-r).C_(n) =""^(2n)C_((n+r)) and hence deduce that Prove that : C_(0)^(2)+C_(1)^(2)+C_(2)^(2)+….+C_(n)^(2)=""^(2n)C_(n)

C_(0)-C_(1)+C_(2)-C_(3)+......+(-1)^(r)C_(r)=((-1)^(r)(n-1)!)/(r!*(n-r-1)!)

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

Prove that "^nC_r+2 ^(n)C_(r-1)+ ^(n)C_(r-2) = ^(n+2)C_r .

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

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