Editing Binomial theorem (section)

{{short description|Algebraic expansion of powers of a binomial}} 
{{CS1 config|mode=cs1}}
{{Image frame|width=215
|content=
<math>
\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}
</math>
|caption=The [[binomial coefficient]] <math>\tbinom{n}{k}</math> appears as the {{mvar|k}}th entry in the {{mvar|n}}th row of [[Pascal's triangle]] (where the top is the 0th row <math>\tbinom{0}{0}</math>). Each entry is the sum of the two above it.}}
In [[elementary algebra]], the '''binomial theorem''' (or '''binomial expansion''') describes the [[Polynomial expansion|algebraic expansion]] of [[exponentiation|powers]] of a [[binomial (polynomial)|binomial]]. According to the theorem, the power {{tmath|\textstyle (x+y)^n}} expands into a [[polynomial]] with terms of the form {{tmath|\textstyle ax^ky^m }}, where the exponents {{tmath|k}} and {{tmath|m}} are [[nonnegative integer]]s satisfying {{tmath|1= k + m = n}} and the [[coefficient]] {{tmath|a}} of each term is a specific [[positive integer]] depending on {{tmath|n}} and {{tmath|k}}.  For example, for {{tmath|1= n = 4}},
<math display=block>(x+y)^4 = x^4 + 4 x^3y + 6 x^2 y^2 + 4 x y^3 + y^4. </math>

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