Calculate nPr (permutations) and nCr (combinations) for any n and r.
nPr = n!/(n-r)! (order matters). nCr = n!/(r!(n-r)!) (order doesn't matter).
Permutations and combinations both count ways to select items from a larger group, and the formulas look similar, but they answer genuinely different questions. A permutation counts arrangements where order matters — nPr = n! ÷ (n−r)! — so choosing a 1st, 2nd and 3rd place winner from 10 racers is a permutation, because swapping who finishes 1st versus 2nd changes the outcome entirely even if the same three people are involved. A combination counts selections where order doesn't matter — nCr = n! ÷ [r! × (n−r)!] — so choosing 3 people from 10 to form a committee is a combination, because the group {Alice, Bob, Carol} is the same committee regardless of which order the three names were picked in.
The practical test worth applying before reaching for either formula: if rearranging the same selected items would count as a genuinely different outcome, it's a permutation; if rearranging them describes the identical outcome, it's a combination. This single distinction explains why permutations always produce a larger count than combinations for the same n and r — a combination formula is exactly the permutation formula divided by r! (the number of ways to reorder the r selected items), since every unique combination corresponds to r! different permutations that all collapse into that same one combination once order stops mattering.
Order. A permutation counts arrangements where the order matters (1st, 2nd, 3rd place from a race) — swapping the order creates a different outcome. A combination counts selections where order doesn't matter (choosing committee members) — the same group is the same outcome regardless of pick order.
Permutations: nPr = n! ÷ (n−r)!. Combinations: nCr = n! ÷ [r! × (n−r)!]. The combination formula is the permutation formula divided by r! — accounting for the fact that each unique combination corresponds to r! different orderings that all count as the same combination.
Ask whether rearranging the same selected items changes the outcome. If yes (assigning 1st/2nd/3rd place, arranging books on a shelf, creating a PIN code), it's a permutation. If no (picking a committee, choosing lottery numbers as a set, selecting a team), it's a combination.
Because a permutation count includes every possible ordering of each selected group, while a combination count treats all those orderings as one single outcome. Since there are r! ways to order any r selected items, the permutation count is always exactly r! times larger than the corresponding combination count.