Combination

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In mathematics a combination is a way of selecting several things out of a larger group, where (unlike permutations) order does not matter. In smaller cases it is possible to count the number of combinations. For example given three fruit, an apple, orange and pear say, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient

which can be written using factorials as \frac{n!}{k!(n-k)!} whenever k\leq n, and which is zero when k > n. The set of all k-combinations of a set S is sometimes denoted by

Combinations can consider the combination of n things taken k at a time without or with repetitions.[1] In the above example repetitions were not allowed. If however it was possible to have two of any one kind of fruit there would be 3 more combinations: one with two apples, one with two oranges, and one with two pears.

With large sets, it becomes necessary to use mathematics to find the number of combinations. For example, a poker hand can be described as a 5-combination (k = 5) of cards from a 52 card deck (n = 52). The 5 cards of the hand are all distinct, and the order of cards in the hand does not matter. There are 2,598,960 such combinations, and the chance of drawing any one hand at random is 1 / 2,598,960.

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Number of k-combinations

The number of k-combinations from a given set S of n elements is often denoted in elementary combinatorics texts by C(nk), or by a variation such as C^n_k, nCk, nCk or even C_n^k (the latter form is standard in French and Polish texts). The same number however occurs in many other mathematical contexts, where it is denoted by \tbinom nk; notably it occurs as coefficient in the binomial formula, hence its name binomial coefficient. One can define \tbinom nk for all natural numbers k at once by the relation

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