Question

Prove that nPr= (n-1)Pr+r.(n-1)P(r-1)

Answer 1

Hello mate here is your answer.

nPr=n!(n−r)!=n(n−1)!(n−r)(n−r−1)!=(n−r+r)(n−r).(n−1)!(n−r−1)!=(1+rn−r).(n−1)!(n−r−1)!=(n−1)!(n−r−1)!+(rn−r)(n−1)!(n−r−1)!=(n−1)!(n−1−r)!+r{(n−1)!(n−r).(n−r−1)!}=n−1Pr+r{(n−1)!(n−r)!}=n−1Pr+r{(n−1)!(n−1−(r−1))!}⇒nPr=n−1Pr+r.n−1Pr−1 (Hence proved)

Hope it helps you.