Permutations (nPr)

A permutation is a selection where order matters: gold, silver and bronze are three different results, not one. With ten sprinters in a final there are 10P3 = 10!/7! = 10 × 9 × 8 = 720 possible podiums. The formula is just the falling product n(n−1)…(n−r+1) written as a ratio of factorials, since dividing by (n − r)! cancels the tail you never reach.

Note that there is no closed-form way to run this backwards, so leaving n or r blank finds it numerically, through the gamma function that extends the factorial between whole numbers. The classic trap is choosing permutations where combinations belong: a five-card poker hand is not 52P5 = 311,875,200, because the same five cards dealt in a different order are the same hand. Divide by 5! to get 2,598,960. Ask yourself whether rearranging the chosen items produces a genuinely different outcome. If yes, permutations; if no, combinations. A four-letter code from 26 distinct letters with no repeats allows 26P4 = 26 × 25 × 24 × 23 = 358,800 codes.