Factorial of 284

Counting with 284!

There are 1.0681 × 10^575 different ways to arrange 284 different books on a shelf: 284 choices for the first place, 283 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 284 runners, there are 284 × 283 × 282 = 22,664,904 possible podiums. If the order does not matter, divide by 3! = 6 to get 3,777,484 ways to choose a group of three. Choosing a pair from 284 people can be done in 40,186 ways.

In probability, if 284 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 1.0681 × 10^575. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 284!?

284! has 576 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 2241, 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 284!

Every prime up to 284 divides 284!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 284! that gives 2^280 × 3^139 × 5^69 × 7^45 × 11^27 × 13^22 × 17^16 × 19^14 × 23^12 × 29^9 × … (61 primes in all). The 69 trailing zeros come from pairing the 69 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 284 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 284! ÷ e, so about 36.8% of all 1.0681 × 10^575 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 284! yourself

In Excel or Google Sheets, type =FACT(284). 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(284) returns the exact integer with all 576 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.