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! ≈ 1024 and 25! ≈ 1025.
Said another way, the curves for n! and 10n cross approximately midway between 24 and 25. To the left of the crossing, n! < 10n and to the right of the crossing n! > 10n.
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 1026.
Left of the cut
Suppose you want to estimate n! for n < 24. You know n! < 10n, but maybe you’d like to be a little more precise.
I’ll suppose you know n! for n up to 6. The approximation
log10 n! ≈ n − 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
log10 n! ≈ 7n/4 − 20
has an absolute error less than 3.
Note that calculating 7n/4 as n + n/2 + n/4 is probably easier than calculating (7n)/4.
I recall that 69! is about 10^98 from playing with calculators long ago. Early scientific calculators that first had the factorial function (such as the TI SR-50) would only display a value with a two-digit power of 10 (up to 10^99), and 69! was the largest factorial that would fit in that range.