Factorial/Examples/1000

Example of Factorial
The factorial of $1\,000$ starts:
 * $402,387,260,077 \ldots$

and has $2568$ digits, of which the last $249$ are $0$.

Proof
From Stirling's Formula:
 * $n! \sim \sqrt {2 \pi n} \left({\dfrac n e}\right)^n$

whence an approximate value for $1\,000!$ can be calculated.

Let $d$ be the number of digits in $1\,000!$

From Number of Digits in Factorial:
 * $d = 1 + \left\lfloor{\left({n + \dfrac 1 2}\right) \log_{10} n - 0.43429 \ 4481 \, n + 0.39908 \ 9934}\right\rfloor$

from which the result can be calculated by setting $n = 1000$.

From Prime Factors of $1000!$:
 * the multiplicity of $5$ in $1000!$ is $249$
 * the multiplicity of $2$ in $1000!$ is $994$.

Therefore there is a factor of $10^{249}$ in $1000!$, but not $10^{250}$.

Hence there are $249$ instances of $0$ at the end of $1000!$.