Binomial theorem

${\displaystyle {\begin{array}{c}1\\1\quad 1\\1\quad 2\quad 1\\1\quad 3\quad 3\quad 1\\1\quad 4\quad 6\quad 4\quad 1\\1\quad 5\quad 10\quad 10\quad 5\quad 1\\1\quad 6\quad 15\quad 20\quad 15\quad 6\quad 1\\1\quad 7\quad 21\quad 35\quad 35\quad 21\quad 7\quad 1\end{array}}}$

The binomial coefficient ${\displaystyle {\tbinom {n}{k}}}$ appears as the kth entry in the nth row of Pascal's triangle (where the top is the 0th row ${\displaystyle {\tbinom {0}{0}}}$). Each entry is the sum of the two above it.

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power ⁠$\textstyle (x+y)^{n}$⁠ expands into a polynomial with terms of the form ⁠$\textstyle ax^{k}y^{m}$⁠, where the exponents ⁠$k$⁠ and ⁠$m$⁠ are nonnegative integers satisfying ⁠$k+m=n$⁠ and the coefficient ⁠$a$⁠ of each term is a specific positive integer depending on ⁠$n$⁠ and ⁠$k$⁠. For example, for ⁠$n=4$⁠, $${\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}$$

The coefficient ⁠$a$⁠ in each term ⁠$\textstyle ax^{k}y^{m}$⁠ is known as the binomial coefficient ⁠${\tbinom {n}{k}}$⁠ or ⁠${\tbinom {n}{m}}$⁠ (the two have the same value). These coefficients for varying ⁠$n$⁠ and ⁠$k$⁠ can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where ⁠${\tbinom {n}{k}}$⁠ gives the number of different combinations (i.e. subsets) of ⁠$k$⁠ elements that can be chosen from an ⁠$n$⁠-element set. Therefore ⁠${\tbinom {n}{k}}$⁠ is usually pronounced as "⁠$n$⁠ choose ⁠$k$⁠".