Factorial of 476

Counting with 476!

There are 3.5867 × 10^1069 different ways to arrange 476 different books on a shelf: 476 choices for the first place, 475 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 476 runners, there are 476 × 475 × 474 = 107,171,400 possible podiums. If the order does not matter, divide by 3! = 6 to get 17,861,900 ways to choose a group of three. Choosing a pair from 476 people can be done in 113,050 ways.

In probability, if 476 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 3.5867 × 10^1069. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 476!?

476! has 1070 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 4383, 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 476!

Every prime up to 476 divides 476!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 476! that gives 2^470 × 3^233 × 5^117 × 7^78 × 11^46 × 13^38 × 17^29 × 19^26 × 23^20 × 29^16 × … (91 primes in all). The 117 trailing zeros come from pairing the 117 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 476 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 476! ÷ e, so about 36.8% of all 3.5867 × 10^1069 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 476! yourself

In Excel or Google Sheets, type =FACT(476). 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(476) returns the exact integer with all 1070 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.