Factorial of 398

Counting with 398!

There are 4.0121 × 10^863 different ways to arrange 398 different books on a shelf: 398 choices for the first place, 397 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 398 runners, there are 398 × 397 × 396 = 62,570,376 possible podiums. If the order does not matter, divide by 3! = 6 to get 10,428,396 ways to choose a group of three. Choosing a pair from 398 people can be done in 79,003 ways.

In probability, if 398 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 4.0121 × 10^863. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 398!?

398! has 864 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 3528, 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 398!

Every prime up to 398 divides 398!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 398! that gives 2^393 × 3^195 × 5^97 × 7^65 × 11^39 × 13^32 × 17^24 × 19^21 × 23^17 × 29^13 × … (78 primes in all). The 97 trailing zeros come from pairing the 97 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 398 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 398! ÷ e, so about 36.8% of all 4.0121 × 10^863 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 398! yourself

In Excel or Google Sheets, type =FACT(398). 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(398) returns the exact integer with all 864 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.