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Fungrim entry: 4e7120

(nk)≤nnkk(n−k)n−k{n \choose k} \le \frac{{n}^{n}}{{k}^{k} {\left(n - k\right)}^{n - k}}
Assumptions:n∈Z≥0  and⁡  k∈{0,1,…,n}n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; k \in \{0, 1, \ldots, n\}
TeX:
{n \choose k} \le \frac{{n}^{n}}{{k}^{k} {\left(n - k\right)}^{n - k}}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; k \in \{0, 1, \ldots, n\}
Definitions:
Fungrim symbol Notation Short description
Binomial(nk){n \choose k} Binomial coefficient
Powab{a}^{b} Power
ZZGreaterEqualZ≥n\mathbb{Z}_{\ge n} Integers greater than or equal to n
Range{a,a+1,…,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
Source code for this entry:
Entry(ID("4e7120"),
    Formula(LessEqual(Binomial(n, k), Div(Pow(n, n), Mul(Pow(k, k), Pow(Sub(n, k), Sub(n, k)))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(k, Range(0, n)))))

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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