Editing Binomial theorem (section)

== Statement ==
According to the theorem, the expansion of any nonnegative integer power {{mvar|n}} of the binomial {{math|''x'' + ''y''}} is a sum of the form
<math display="block">(x+y)^n = {n \choose 0}x^n y^0 + {n \choose 1}x^{n-1} y^1 + {n \choose 2}x^{n-2} y^2 + \cdots + {n \choose n}x^0 y^n,</math>
where each <math> \tbinom nk </math> is a positive integer known as a [[binomial coefficient]], defined as

<math display=block>\binom nk = \frac{n!}{k!\,(n-k)!} = \frac{n(n-1)(n-2)\cdots(n-k + 1)}{k(k-1)(k-2)\cdots2\cdot1}.</math>

This formula is also referred to as the '''binomial formula''' or the '''binomial identity'''. Using [[Capital-sigma notation|summation notation]], it can be written more concisely as
<math display="block">(x+y)^n = \sum_{k=0}^n {n \choose k}x^{n-k}y^k = \sum_{k=0}^n {n \choose k}x^{k}y^{n-k}.</math>

The final expression follows from the previous one by the symmetry of {{mvar|x}} and {{mvar|y}} in the first expression, and by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical, <math display=inline>\binom nk = \binom n{n-k}.</math>

A simple variant of the binomial formula is obtained by [[substitution (algebra)|substituting]] {{math|1}} for {{mvar|y}}, so that it involves only a single [[Variable (mathematics)|variable]]. In this form, the formula reads
<math display=block>\begin{align}
(x+1)^n
&= {n \choose 0}x^0 + {n \choose 1}x^1 + {n \choose 2}x^2 + \cdots + {n \choose n}x^n \\[4mu]
&= \sum_{k=0}^n {n \choose k}x^k. \vphantom{\Bigg)}
\end{align}</math><!-- \vphantom{\Bigg)} works around a mediawiki scrollbar bug -->