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Fungrim entry: 6f7746

∑k=0n(nk)xkyn−k=(x+y)n\sum_{k=0}^{n} {n \choose k} {x}^{k} {y}^{n - k} = {\left(x + y\right)}^{n}
Assumptions:x∈C and⁡ y∈C and⁡ n∈Z≥0x \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
Binomial(nk){n \choose k} Binomial coefficient
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZ≥n\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))), Tuple(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.

2019-06-18 07:49:59.356594 UTC