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

Factorials and binomial coefficients