# Fungrim entry: d3baaf

${n \choose k} < \frac{1}{\sqrt{2 \pi}} \sqrt{\frac{n}{k \left(n - k\right)}} \frac{{n}^{n}}{{k}^{k} {\left(n - k\right)}^{n - k}}$
Assumptions:$n \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, n - 1\}$
TeX:
{n \choose k} < \frac{1}{\sqrt{2 \pi}} \sqrt{\frac{n}{k \left(n - k\right)}} \frac{{n}^{n}}{{k}^{k} {\left(n - k\right)}^{n - k}}

n \in \mathbb{Z}_{\ge 2} \;\mathbin{\operatorname{and}}\; k \in \{1, 2, \ldots, n - 1\}
Definitions:
Fungrim symbol Notation Short description
Binomial${n \choose k}$ Binomial coefficient
Sqrt$\sqrt{z}$ Principal square root
Pi$\pi$ The constant pi (3.14...)
Pow${a}^{b}$ Power
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Range$\{a, a + 1, \ldots, b\}$ Integers between given endpoints
Source code for this entry:
Entry(ID("d3baaf"),
Formula(Less(Binomial(n, k), Mul(Mul(Div(1, Sqrt(Mul(2, Pi))), Sqrt(Div(n, Mul(k, Sub(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(2)), Element(k, Range(1, Sub(n, 1))))))

## 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