Factorial of 209

Counting with 209!

There are 5.0392 × 10^395 different ways to arrange 209 different books on a shelf: 209 choices for the first place, 208 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 209 runners, there are 209 × 208 × 207 = 8,998,704 possible podiums. If the order does not matter, divide by 3! = 6 to get 1,499,784 ways to choose a group of three. Choosing a pair from 209 people can be done in 21,736 ways.

In probability, if 209 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 5.0392 × 10^395. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 209!?

209! has 396 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 1530, 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 209!

Every prime up to 209 divides 209!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 209! that gives 2^205 × 3^101 × 5^50 × 7^33 × 11^20 × 13^17 × 17^12 × 19^11 × 23^9 × 29^7 × … (46 primes in all). The 50 trailing zeros come from pairing the 50 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 209 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 209! ÷ e, so about 36.8% of all 5.0392 × 10^395 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 209! yourself

In Excel or Google Sheets, type =FACT(209). 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(209) returns the exact integer with all 396 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.