Binomial coefficient

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In mathematics, the binomial coefficient  \tbinom nk is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n.

In combinatorics, \tbinom nk is interpreted as the number of k-element subsets (the k-combinations) of an n-element set, that is the number of ways that k things can be "chosen" from a set of n things. Hence, \tbinom nk is often read as "n choose k" and is called the choose function of n and k.

The notation \tbinom nk was introduced by Andreas von Ettingshausen in 1826,[1] although the numbers were already known centuries before that (see Pascal's triangle). The earliest known detailed discussion of binomial coefficients is in a tenth-century commentary, due to Halayudha, on an ancient Hindu classic, Pingala's chandaḥśāstra. In about 1150, the Hindu mathematician Bhaskaracharya gave a very clear exposition of binomial coefficients in his book Lilavati.[2]

Alternative notations include C(n, k), nCk, nCk, \scriptstyle C^{n}_{k}, \scriptstyle C^{k}_{n},[3] in all of which the C stands for combinations or choices.

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