Count the ways to choose r items from n, with and without order and with and without repetition: nCr, nPr, nʳ and multisets.
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With total items (n) 52, items chosen (r) 5, the combinations (order does not matter) is 2,598,960.
| Total items (n) | 52 |
|---|---|
| Items chosen (r) | 5 |
| Combinations (order does not matter) | 2,598,960 |
|---|---|
| Permutations (order matters) | 311,875,200 |
| Permutations with repetition | 380,204,032 |
| Combinations with repetition | 3,819,816 |
Use combinations when order does not matter, such as lottery numbers or a hand of cards, and permutations when it does, such as race finishing positions or a PIN. Repetition means the same item can be chosen more than once. All four counts are calculated exactly with whole-number arithmetic; very large results are summarised in scientific notation.
nPr = n! ÷ (n − r)!. nCr = n! ÷ (r!(n − r)!). With repetition: nʳ ordered, C(n + r − 1, r) unordered. Use combinations when order does not matter, such as lottery numbers or a hand of cards, and permutations when it does, such as race finishing positions or a PIN. Repetition means the same item can be chosen more than once. All four counts are calculated exactly with whole-number arithmetic; very large results are summarised in scientific notation.
With total items (n) 52, items chosen (r) 5, the combinations (order does not matter) is 2,598,960. Permutations (order matters): 311,875,200; Permutations with repetition: 380,204,032; Combinations with repetition: 3,819,816.
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52C5 = 2,598,960. Order does not matter in a hand, so this is a combination.
A permutation counts arrangements, so ABC and CBA are different. A combination counts selections, so they are the same. For the same n and r, nPr = nCr × r!.
Calculate the combinations for the draw, for example 6 numbers from 49 gives 49C6 = 13,983,816. One ticket's chance of matching all six is 1 in that number.