Editing Binomial coefficient (section)

{{short description|Number of subsets of a given size}}
{{redirect|nCk||NCK (disambiguation)}}
[[Image:Pascal's triangle 5.svg|right|thumb|200px|The binomial coefficients can be arranged to form [[Pascal's triangle]], in which each entry is the sum of the two immediately above.]]
[[Image:binomial_theorem_visualisation.svg|thumb|300px|Visualisation of binomial expansion up to the [[4th power]]]]

In [[mathematics]], the '''binomial coefficients''' are the positive [[integer]]s that occur as [[coefficient]]s in the [[binomial theorem]]. Commonly, a binomial coefficient is indexed by a pair of integers {{math|''n'' ≥ ''k'' ≥ 0}} and is written <math>\tbinom{n}{k}.</math> It is the coefficient of the {{math|''x''<sup>''k''</sup>}} term in the [[polynomial expansion]] of the [[binomial (polynomial)|binomial]] [[exponentiation|power]] {{math|(1 + ''x'')<sup>''n''</sup>}}; this coefficient can be computed by the multiplicative formula
: <math>\binom nk = \frac{n\times(n-1)\times\cdots\times(n-k+1)}{k\times(k-1)\times\cdots\times1},</math>
which using [[factorial]] notation can be compactly expressed as
: <math>\binom{n}{k} = \frac{n!}{k! (n-k)!}.</math>

For example, the fourth power of {{math|1 + ''x''}} is
: <math>\begin{align}
(1 + x)^4 &= \tbinom{4}{0} x^0 + \tbinom{4}{1} x^1 + \tbinom{4}{2} x^2 + \tbinom{4}{3} x^3 + \tbinom{4}{4} x^4 \\
&= 1 + 4x + 6 x^2 + 4x^3 + x^4,
\end{align}</math>
and the binomial coefficient <math>\tbinom{4}{2} =\tfrac{4\times 3}{2\times1} = \tfrac{4!}{2!2!} = 6</math> is the coefficient of the {{math|''x''<sup>2</sup>}} term.

Arranging the numbers <math>\tbinom{n}{0}, \tbinom{n}{1}, \ldots, \tbinom{n}{n}</math> in successive rows for {{math|1= ''n'' = 0, 1, 2, ...}} gives a triangular array called [[Pascal's triangle]], satisfying the [[recurrence relation]]
: <math>\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k} .</math>

The binomial coefficients occur in many areas of mathematics, and especially in [[combinatorics]]. In combinatorics the symbol <math>\tbinom{n}{k}</math> is usually read as "{{mvar|n}} choose {{mvar|k}}" because there are <math>\tbinom{n}{k}</math> ways to choose an (unordered) subset of {{mvar|k}} elements from a fixed set of {{mvar|n}} elements. For example, there are <math>\tbinom{4}{2}=6</math> ways to choose {{math|2}} elements from {{math|{{mset|1, 2, 3, 4}}}}, namely {{math|{{mset|1, 2}}}}, {{math|{{mset|1, 3}}}}, {{math|{{mset|1, 4}}}}, {{math|{{mset|2, 3}}}}, {{math|{{mset|2, 4}}}} and {{math|{{mset|3, 4}}}}.

The first form of the binomial coefficients can be generalized to <math>\tbinom{z}{k}</math> for any [[complex number]] {{mvar|z}} and integer {{math|''k'' ≥ 0}}, and many of their properties continue to hold in this more general form.