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

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