math - How to find nPr (permutations) efficiently? -

is there better way using basic formula n!/(n-r)! have ncr(combinations) ncr = (n-l)cr + (n-1)c(r-1) ?

how this: npr = (n−1)pr + (n−1)p(r−1) ⋅ r

rationale: npr denotes number of ways choose r elements n while noting order , not putting them back. in above recursion distinguish 2 cases. either don't choose nth element, in case you'll choosing r elements set of (n−1). or you'll choosing nth element well, in case you'll choosing other (r−1) elements set of (n−1), , there r possibilities @ point in order chose nth element.

apart this, note can avoid 2 factorials taking product on difference:

  n ─┬──┬─       n!  │  │   = ──── = (n−r+1)⋅(n−r+2)⋅…⋅(n−1)⋅n = npr  │  │        r! i=n−r+1 

this leads yet recursive formulation: npr = (n−1)p(r−1) ⋅ n