Combination calculator
Choosing five cards from fifty-two where order does not matter gives 2,598,960 possible hands. If order did matter it would be 311,875,200 — exactly 120 times more, because five cards can be arranged in 5! = 120 ways. That factor of r! is the entire difference between a combination and a permutation, and it is why nCr is always the smaller number.
A combination counts selections where order does not matter: C(52,5) = 2,598,960 possible poker hands. A permutation counts arrangements where it does: P(52,5) = 311,875,200 — exactly 5! times more.
How to count combinations
The four numbers here cover every counting situation, and picking the right one comes down to two questions. Does order matter? And can an item be chosen more than once? A lottery draw is combinations without repetition. A four-digit PIN is arrangements with repetition, which is why there are 10,000 of them rather than 5,040. A podium finish is permutations without repetition. Getting the wrong one is by far the most common error in counting problems.
Questions
Order. Combinations treat ABC and CBA as the same selection; permutations count them separately.