Factorial of 319

Counting with 319!

There are 6.6128 × 10^661 different ways to arrange 319 different books on a shelf: 319 choices for the first place, 318 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 319 runners, there are 319 × 318 × 317 = 32,157,114 possible podiums. If the order does not matter, divide by 3! = 6 to get 5,359,519 ways to choose a group of three. Choosing a pair from 319 people can be done in 50,721 ways.

In probability, if 319 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 6.6128 × 10^661. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 319!?

319! has 662 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 2583, 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 319!

Every prime up to 319 divides 319!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 319! that gives 2^312 × 3^156 × 5^77 × 7^51 × 11^31 × 13^25 × 17^19 × 19^16 × 23^13 × 29^11 × … (66 primes in all). The 77 trailing zeros come from pairing the 77 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 319 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 319! ÷ e, so about 36.8% of all 6.6128 × 10^661 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 319! yourself

In Excel or Google Sheets, type =FACT(319). 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(319) returns the exact integer with all 662 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.