Prove that `(n-r+1)"^nC_(r-1) `= `n"^(n-1)C_(r-1)`

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

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.

Prove that "^nC_r ^rC_5= ^nC_5 ^(n-5)C_(r-5) .

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 sum_(r=0)^n 3^r "^nC_r= 4^n .

Verify that ""^nC_r= frac (n)(r) "^(n-1)C_(r-1) and hence prove that "^nC_r= (n!)/(r!(n-r)!) .

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

If ""^(n)C_(r ): ""^(n)C_(r+1)=1:2 and ""^(n)C_(r+1): ""^(n)C_(r+2)=2:3 , find n and r.

Prove that "^(n-1)C_3+ ^(n-1)C_4 > ^nC_3 if n >7.

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