Factorial of 312

Counting with 312!

There are 2.1020 × 10^644 different ways to arrange 312 different books on a shelf: 312 choices for the first place, 311 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 312 runners, there are 312 × 311 × 310 = 30,079,920 possible podiums. If the order does not matter, divide by 3! = 6 to get 5,013,320 ways to choose a group of three. Choosing a pair from 312 people can be done in 48,516 ways.

In probability, if 312 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 2.1020 × 10^644. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 312!?

312! has 645 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 2565, 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 312!

Every prime up to 312 divides 312!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 312! that gives 2^308 × 3^153 × 5^76 × 7^50 × 11^30 × 13^25 × 17^19 × 19^16 × 23^13 × 29^10 × … (64 primes in all). The 76 trailing zeros come from pairing the 76 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 312 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 312! ÷ e, so about 36.8% of all 2.1020 × 10^644 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 312! yourself

In Excel or Google Sheets, type =FACT(312). 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(312) returns the exact integer with all 645 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.