Factorial of 383

Counting with 383!

There are 5.2617 × 10^824 different ways to arrange 383 different books on a shelf: 383 choices for the first place, 382 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 383 runners, there are 383 × 382 × 381 = 55,742,586 possible podiums. If the order does not matter, divide by 3! = 6 to get 9,290,431 ways to choose a group of three. Choosing a pair from 383 people can be done in 73,153 ways.

In probability, if 383 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 5.2617 × 10^824. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 383!?

383! has 825 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 3357, 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 383!

Every prime up to 383 divides 383!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 383! that gives 2^375 × 3^188 × 5^94 × 7^62 × 11^37 × 13^31 × 17^23 × 19^21 × 23^16 × 29^13 × … (76 primes in all). The 94 trailing zeros come from pairing the 94 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 383 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 383! ÷ e, so about 36.8% of all 5.2617 × 10^824 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 383! yourself

In Excel or Google Sheets, type =FACT(383). 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(383) returns the exact integer with all 825 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.