Permutations, k-Permutations and Combinations

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Number of Permutations

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The number of ways to arrange different objects in a row is . The exclamation mark "!" is read as "factorial". Of course, the product is the same in reverse order: . Each such arrangement is called a permutation. For consistency, it is assumed that .

Number of k-Permutations

If only of the objects are to be arranged in a row, the formula is

,

with factors. If , . Such an arrangement is called a partial permutation, or a -permutation. Clearly , because all objects are being arranged; the formula reduces to because the denominator is .

Number of Combinations

The number of ways to choose a subset of objects from objects is

.

Therefore, . Each choice of a subset is called a combination. Another notation for is . Again, if , . A special case is .

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Contributed by: George Beck (May 2018)
Open content licensed under CC BY-NC-SA


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