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→Sums of the binomial coefficients: repeated self-promotion, sockpuppeteering |
provide basic multiplicative formula in lede; the factorial formula is just a condensation of it |
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[[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>}}
:<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>
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&= 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>n=0,1,2,\ldots</math> gives a triangular array called [[Pascal's triangle]], satisfying the [[recurrence relation]]
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