Suppose you’d like to have a very rough idea how large *n*! is for, say, *n* less than 100.

If you’re familiar with such things, your Pavlovian response to “factorial” and “approximation” is Stirling’s approximation. Although Stirling’s approximation is extremely useful—I probably use it every few weeks—it is not conducive to mental calculation.

## The cut point

It’s useful to know that 24! ≈ 10^{24} and 25! ≈ 10^{25}.

Said another way, the curves for *n*! and 10^{n} cross approximately midway between 24 and 25. To the left of the crossing, *n*! < 10^{n} and to the right of the crossing *n*! > 10^{n}.

So, for example, if you hear someone refer to permutations of the English alphabet, you know the number of permutations is going to be north of 10^{26}.

## Left of the cut

Suppose you want to estimate *n*! for *n* < 24. You know *n*! < 10^{n}, but maybe you’d like to be a little more precise.

I’ll suppose you know *n*! for *n* up to 6. The approximation

log_{10} *n*! ≈ *x* − 2

has an absolute error of less than 1.5 for *n* = 7, 8, 9, …, 23.

## Right of cut

For *n* = 26, 27, 28, …, 100 the approximation

log_{10} *n*! ≈ 7*n*/4 − 20

has an absolute error less than 3.

## Related posts

The post Mentally approximating factorials first appeared on John D. Cook.