Enter how many items there are and how many you pick, and get the number of permutations (order matters) and combinations (order does not).
Ask yourself: would swapping two of the chosen items give a different result? If yes, order matters, and you want permutations. If no, you want combinations. Picking a gold, silver and bronze medallist from 8 runners is a permutation, because the same three people in a different order is a different outcome. Picking 3 of the 8 to form a committee is a combination, because the group is the same whatever order you name them in.
Permutations, nPr = n! ÷ (n − r)!. Combinations, nCr = n! ÷ (r! × (n − r)!). Every combination can be arranged in r! different orders, which is why nCr is nPr divided by r!. For the medal example, 8P3 = 8 × 7 × 6 = 336 ways to award the medals, while 8C3 = 336 ÷ 6 = 56 possible committees.
The number of five-card poker hands from a 52-card deck is 52C5 = 2,598,960. A 6-number lottery drawn from 49 balls has 49C6 = 13,983,816 possible tickets, so one ticket has a 1 in about 14 million chance of the jackpot. If you want to turn counts into odds, the Probability Calculator picks up where this one leaves off: the chance of a specific outcome is 1 divided by the number of equally likely possibilities.
The formulas above assume each item can be used once. A 4-digit PIN is different, since digits can repeat, so there are 10 × 10 × 10 × 10 = 104 = 10,000 arrangements. That is the "with repetition" line in the results: nr for ordered picks, and C(n + r − 1, r) for unordered picks, such as choosing 3 scoops from 6 ice cream flavours where you can repeat a flavour (56 ways). Choose the "with repetition" figure whenever the same item can come up more than once.
Factorials grow extremely fast. 20! is already over 2 quintillion, and 100! has 158 digits. For answers up to a few thousand digits the calculator gives you the exact whole number. Beyond that, it shows the size in scientific notation, since nobody can use a 300,000-digit number in practice. If you are working with sets that big, you are usually after a probability, not the raw count.
Counting possibilities is usually step one. To turn a count into the odds of an event, use the Probability Calculator. If your data is a list of numbers rather than a counting problem, the Statistics Calculator summarises it. And for questions about how many people you need to survey, rather than how many ways something can happen, the Sample Size Calculator is the right tool.
In a permutation the order of the chosen items matters. In a combination it does not. For the same n and r there are always at least as many permutations as combinations.
Use n! ÷ (r! × (n − r)!). For 6C2 that is 720 ÷ (2 × 24) = 15.
0! is defined as 1. That is why nC0 and nCn are both 1: there is exactly one way to choose nothing, and one way to choose everything.
Not when items cannot repeat, because you cannot pick more items than exist. With repetition allowed, r can be larger than n, and the n^r formula still applies.
There are 2,598,960 hands, which is 52C5.
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