Fungrim entry: d3baaf

{n \choose k} \lt \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} \lt \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
`ConstPi`	$\pi$	The constant pi (3.14...)
`Pow`	${a}^{b}$	Power
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZBetween`	$\{a, a + 1, \ldots b\}$	Integers between a and b inclusive

Source code for this entry:

Entry(ID("d3baaf"),
    Formula(Less(Binomial(n, k), Mul(Mul(Div(1, Sqrt(Mul(2, ConstPi))), 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, ZZBetween(1, Sub(n, 1))))))

Topics using this entry

Factorials and binomial coefficients