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Combinatorics

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Permutations without repetition

The k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set S of n unique elements. When k is equal to the size n of the set, these are the permutations of the set and their number equals n! (n factorial). If k < n, the number of all possible k-permutations is:
 nPk = n! (n - k)!

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Permutations with repetition

Ordered arrangements of length k of the elements from a set S where the same element may appear more than once are called k-tuples, but have sometimes been referred to as permutations with repetition. They are also called words over the alphabet S in some contexts. If the set S has n elements, the number of k-tuples over S is:
 Pk(n) = nk

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Combinations without repetition

A combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter. If the set has n elements, the number of k-combinations (subsets with k elements) is:
 nCk = n! k!(n - k)! = ( n k )

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Combinations with repetition

A k-combination with repetition, or multisubset of size k from a set S is given by a sequence of k elements of S, where the same element may appear more than once and order is irrelevant. If the set has n elements, the number of k-combinations with repetitions is:
 Cr(n, k) = (k + n - 1)! k!(n - 1)!

Calculator (fill both fields and click 'Calculate'):

Total number of items:n =

How many to choose:k =

Number of combinations:

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