Prove that C(n,r) = C(n,n-r)

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

Prove that ((n),(r))+2((n),(r-1))+((n),(r-2))=((n+2),(r))

Prove that : ^nP_r= n"^(n-1)P_(r-1) , for all natural numbers n and r for which the symbols are defined.

Prove that : ^nP_r= "^(n-1)P_r+r ^(n-1)P_(r-1) , for all natural numbers n and r for which the symbols are defined.

Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that P^(2)R^(n) = S^(n) .

Prove that : ^(2n)C_n = (2^n [1.3.5. ..........(2n-1)])/(n!) .

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

If C_(0) , C_(1), C_(2), …, C_(n) are the binomial coefficients in the expansion of (1 + x)^(n) , prove that (C_(0) + 2C_(1) + C_(2) )(C_(1) + 2C_(2) + C_(3))…(C_(n-1) + 2C_(n) + C_(n+1)) ((n-2)^(n))/((n+1)!) prod _(r=1)^(n) (C_(r-1) + C_(r)) .

If a,b,c,d are in G.P., prove that (a^(n) + b^(n)), (b^(n) + c^(n)), (c^(n) + d^(n)) are in G.P.

If (1+ x)^(n) = C_(0) + C_(1) x + C_(2)x^(2) + ...+ C_(n)x^(n) , prove that C_(1) + 2C_(2) + 3C_(3) + ...+ n""C_(n) = n*2^(n-1)