Prove that `P(n,r)=nP_r=(n!)/((n-r)!`

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

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

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

11. Prove that nP_(r)=n(n-1)P_(r-1)

Prove that ""^(n)P_(r )= ""^(n)C_(r )*^rP_(r ) .

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

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

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

Prove that : (i) (n!)/(r!)=n(n-1)(n-2)...(r+1) (ii) (n-r+1)*(n!)/((n-r+1)!)=(n!)/((n-r)!) (iii) (n!)/(r!(n-r)!)+(n!)/((r-1)!(n-r+1)!)=((n+1)!)/(r!(n-r+1)!)

If P_(n) is the sum of a G.P. upto n terms (n>=3), then prove that (1-r)(dP_(n))/(dr)=(1-n)P_(n)+nP_(n-1), where r is the common ratio of G.P.