Prove that following identities:
(n+1)P(n,r) = (n-r+1)P(n+1,r)

Prove that: sum_(r=0)^(n) 3^( r) ""^(n)C_(r) = 4^(n) .

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.

If x^p occurs in the expansion of (x^2+1/x)^(2n) , prove that its coefficient is : ((2n)!)/([1/3(4n-p)!][1/3(2n+p)!]) .

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

If t_r is the rth term is the expansion of (1 + a)^n , in ascending powers of a, prove that : r(r+1) t_(r+2)= (n-r+1)(n-r) a^2 t_r .

For the proposition P(n), given by , 1+3+5+.........+(2n-1) = n^2 +2 , prove that P(k) is true implies P(k + 1) is true. But, P(n) is not true for all n in N.

Prove that sum_(r=0)^n^n C_rsinr xcos(n-r)x=2^(n-1)sin(n x)dot

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

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