# Stirling's Formula

In Permutations, we showed that the number of permutations of \(n\) distinct objects is given by the factorial function \(n!\) How quickly does the factorial function \(n!\) grow as a function of \(n\)? This behavior is captured in the approximation known as **Stirling's Formula** (also known as **Stirling's Approximation**).

Stirling's FormulaThe factorial function \(n!\) is approximated by

\[ n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n.\]

Furthermore, for any positive integer \(n\), we have the bounds

\[ \sqrt{2\pi}\ n^{n+1/2}e^{-n} \le n! \le e\ n^{n+1/2}e^{-n}. \]

## Central Binomial Coefficient

The theorem defined in binomial coefficient as \( { 2n \choose n } = \frac { (2n)!} {n!^2} \) for \(n \geq 0 \) and it approaches \( \frac {4^n}{\sqrt{\pi n}} \) asymptotically.

Given the Stirling Formula, we have

\[ n! \approx \sqrt{2 \pi n} \left ( \frac {n}{e} \right )^n \]

\[ (2n)! \approx \sqrt{4 \pi n} \left ( \frac {2n}{e} \right )^{2n} \]

Taking the ratio of the second approximation to the square of the first approximation:

\[ \frac {(2n)!}{n!^2} \rightarrow \sqrt{ \frac {4\pi n}{ (2 \pi n)^2 } } \cdot \frac {(2n)^{2n}}{n^{2n}} \cdot \frac {e^{2n}}{e^{2n}} = \frac {1}{\sqrt{\pi n}} \cdot 2^{2n} = \frac {4^n}{\sqrt{\pi n}} \]

## Catalan Number

Catalan Number occurs many times in various counting problems, it is defined to be \( \large C_n = \frac {1}{n+1} { 2n \choose n } \). For more information, go to Catalan Numbers

**Cite as:**Stirling's Formula.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/stirlings-formula/