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Last update: Saturday 11th of February 2012
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In mathematics, particularly in combinatorics, the binomial coefficient of the natural number n and the integer k is the number of combinations that exist. In other words, given n choices, such as pizza toppings, and k "slots," such as two pizza toppings per pizza, the binomal coefficient will determine how many non-ordered combinations exist, or how many different pizzas one can order.
It is defined to be the natural number
and
where m! denotes the factorial of m. According to Nicholas J. Higham, the
notation was introduced by Albert von Ettinghausen in 1826, although these numbers have been known centuries before that; see Pascal's triangle.
Another name for the binomial coefficient is choose function; the binomial coefficient of n and k is often read as "n choose k". Alternative notations include C(n, k), nCk or 
(C for combination).
The binomial coefficients are the coefficients in the expansion of the binomial (x + y)n (hence the name):
This is generalized by the binomial theorem, which allows the exponent n to be negative or a non-integer. See the article on combination.
The importance of the binomial coefficients (and reason for the alternate name 'choose') lies in the fact that 
is the number of ways that k objects can be chosen from n objects, regardless of order.
In the formula
the numerator gives the number of ways to fill the k slots using the n options, taking into account the order the choices are made. Thus a pizza with mushrooms added before chicken is considered to be different from a pizza with chicken added before mushrooms. The denominator eliminates these repetitions because there are k! ways to rearrange the k slots.
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