# Fungrim entry: 6f7746

$\sum_{k=0}^{n} {n \choose k} {x}^{k} {y}^{n - k} = {\left(x + y\right)}^{n}$
Assumptions:$x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}$
TeX:
\sum_{k=0}^{n} {n \choose k} {x}^{k} {y}^{n - k} = {\left(x + y\right)}^{n}

x \in \mathbb{C} \;\mathbin{\operatorname{and}}\; y \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Sum$\sum_{n} f(n)$ Sum
Binomial${n \choose k}$ Binomial coefficient
Pow${a}^{b}$ Power
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("6f7746"),
Formula(Equal(Sum(Mul(Mul(Binomial(n, k), Pow(x, k)), Pow(y, Sub(n, k))), For(k, 0, n)), Pow(Add(x, y), n))),
Variables(x, y, n),
Assumptions(And(Element(x, CC), Element(y, CC), Element(n, ZZGreaterEqual(0)))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC