Factorial of 388

Counting with 388!

There are 4.5086 × 10^837 different ways to arrange 388 different books on a shelf: 388 choices for the first place, 387 for the second, and so on down to one choice for the last. If you only fill the first three places, as with gold, silver and bronze among 388 runners, there are 388 × 387 × 386 = 57,960,216 possible podiums. If the order does not matter, divide by 3! = 6 to get 9,660,036 ways to choose a group of three. Choosing a pair from 388 people can be done in 75,078 ways.

In probability, if 388 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 4.5086 × 10^837. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 388!?

388! has 838 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 3348, a multiple of 9, as they are for every factorial from 6! on. Stirling's formula, √(2πn) × (n/e)^n, gives a close estimate without multiplying everything out, and gets more accurate as n grows.

Prime factors of 388!

Every prime up to 388 divides 388!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 388! that gives 2^385 × 3^191 × 5^95 × 7^63 × 11^38 × 13^31 × 17^23 × 19^21 × 23^16 × 29^13 × … (76 primes in all). The 95 trailing zeros come from pairing the 95 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 388 items: orders in which no item ends up in its original place, as in a Secret Santa where nobody draws their own name. The count is the whole number nearest to 388! ÷ e, so about 36.8% of all 4.5086 × 10^837 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 388! yourself

In Excel or Google Sheets, type =FACT(388). Spreadsheets keep only 15 significant digits, so from 18! on the result is rounded, and FACT stops at 170!, the largest factorial that fits in a standard floating-point number. In Python, math.factorial(388) returns the exact integer with all 838 digits, because Python integers have no size limit. The gamma function extends factorials to fractions: Γ(n + 1) = n!, so (1/2)! = √π ÷ 2 ≈ 0.8862.

Why is 0! equal to 1?

There is exactly one way to arrange nothing, and the rule n! = n × (n − 1)! only works for n = 1 if 0! = 1. Defining it this way keeps formulas for combinations and series correct at the edges.