Mentally approximating factorials

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.

Related posts

• Approximating (x)
• Approximating 1/(x)
• Mentally computing common functions

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