Binomial theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power ⁠ ( x + y ) n {\displaystyle \textstyle (x+y)^{n}} ⁠ expands into a polynomial with terms of the form ⁠ a x k y m {\displaystyle \textstyle ax^{k}y^{m}} ⁠, where the exponents ⁠ k {\displaystyle k} ⁠ and ⁠ m {\displaystyle m} ⁠ are nonnegative integers satisfying ⁠ k + m = n {\displaystyle k+m=n} ⁠ and the coefficient ⁠ a {\displaystyle a} ⁠ of each term is a specific positive integer depending on ⁠ n {\displaystyle n} ⁠ and ⁠ k {\displaystyle k} ⁠. For example, for ⁠ n = 4 {\displaystyle n=4} ⁠, ( x + y ) 4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 x y 3 + y 4 . {\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.} The coefficient ⁠ a {\displaystyle a} ⁠ in each term ⁠ a x k y m {\displaystyle \textstyle ax^{k}y^{m}} ⁠ is known as the binomial coefficient ⁠ ( n k ) {\displaystyle {\tbinom {n}{k}}} ⁠ or ⁠ ( n m ) {\displaystyle {\tbinom {n}{m}}} ⁠ (the two have the same value). These coefficients for varying ⁠ n {\displaystyle n} ⁠ and ⁠ k {\displaystyle k} ⁠ can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where ⁠ ( n k ) {\displaystyle {\tbinom {n}{k}}} ⁠ gives the number of different combinations (i.e. subsets) of ⁠ k {\displaystyle k} ⁠ elements that can be chosen from an ⁠ n {\displaystyle n} ⁠-element set. Therefore ⁠ ( n k ) {\displaystyle {\tbinom {n}{k}}} ⁠ is usually pronounced as "⁠ n {\displaystyle n} ⁠ choose ⁠ k {\displaystyle k} ⁠".