Three runners finishing first, second and third is a different question from three runners making the final. Both start with the same pool and pick the same number, but one counts orderings and the other counts groups — and the first answer is always the larger, by exactly the number of ways the chosen items can be shuffled among themselves. This page shows both at once so the choice between them is visible rather than assumed.

Using it in three steps

1. 1 Enter n - how many things there are to choose from.
2. 2 Enter r - how many of them you are taking.
3. 3 Read whichever line matches your question: order matters, or it does not.

The one question that decides which formula

Ask whether swapping two of the chosen items produces a different outcome. Gold and silver swapped is a different result, so podium places are permutations. Two lottery numbers swapped is the same ticket, so lottery draws are combinations. Everything else follows from that one test, and getting it wrong is the single most common mistake in this corner of mathematics — usually in the direction of counting each group several times over.

Why these numbers get enormous so fast

Permutations grow faster than almost anything else in ordinary arithmetic. Arranging ten items has 3,628,800 orderings; arranging fifteen has over a trillion. That growth is why a padlock with four dials feels secure and why shuffling a deck of cards reliably produces an arrangement no one has ever seen. It is also why exact answers stop being possible on a screen quite quickly, and the tool says so rather than printing digits it cannot vouch for.