Show that nPr - nP(r-1) = (n-r)*nP (r-1)

Question

Show that _nP_r - _nP_r-1 = (n - r)_nP_r-1

Answer 1

nPr = n!/(n-r)!

nP(r-1) = n!./(n-(r-1))

nPr - nP(r-1) = n!/(n-r)!-n!/(n-(r-1))! = n!/(n-r)!-n!/((n-r)!(n-(r-1))

Get common denominator

(n!(n-(r-1))-n!)/((n-r)!(n-(r-1)) = n!(n-r+1-1)/(n-(r-1))! = (n-r)n!/(n-(r-1))! = (n-r)•nP(r-1)