In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial \((x + y)^n\) into a sum involving terms of the form \(ax^by^c\), where the exponents \(b\) and \(c\) are nonnegative integers with \(b + c = n\), and the coefficient \(a\) of each term is a specific positive integer depending on \(n\) and \(b\). For example, for \(n = 4\),
\((x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}\)
The coefficient a in the term of \(ax^by^c\) is known as the binomial coefficient [Binomial coefficient] \(\binom{n}{b}\) or \(\tbinom {n}{c}}\) (the two have the same value).
$$
\begin{align} (x+y)^{n} &=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k} \\ &=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k} \end{align}
$$