Factorial of 218

Counting with 218!

There are 4.7402 × 10^416 different ways to arrange 218 different books on a shelf: 218 choices for the first place, 217 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 218 runners, there are 218 × 217 × 216 = 10,218,096 possible podiums. If the order does not matter, divide by 3! = 6 to get 1,703,016 ways to choose a group of three. Choosing a pair from 218 people can be done in 23,653 ways.

In probability, if 218 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 4.7402 × 10^416. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 218!?

218! has 417 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 1665, 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 218!

Every prime up to 218 divides 218!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 218! that gives 2^213 × 3^106 × 5^52 × 7^35 × 11^20 × 13^17 × 17^12 × 19^11 × 23^9 × 29^7 × … (47 primes in all). The 52 trailing zeros come from pairing the 52 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 218 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 218! ÷ e, so about 36.8% of all 4.7402 × 10^416 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 218! yourself

In Excel or Google Sheets, type =FACT(218). 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(218) returns the exact integer with all 417 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.