Factorial of 221

Counting with 221!

There are 5.0473 × 10^423 different ways to arrange 221 different books on a shelf: 221 choices for the first place, 220 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 221 runners, there are 221 × 220 × 219 = 10,647,780 possible podiums. If the order does not matter, divide by 3! = 6 to get 1,774,630 ways to choose a group of three. Choosing a pair from 221 people can be done in 24,310 ways.

In probability, if 221 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 5.0473 × 10^423. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 221!?

221! has 424 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 1692, 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 221!

Every prime up to 221 divides 221!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 221! that gives 2^215 × 3^107 × 5^53 × 7^35 × 11^21 × 13^18 × 17^13 × 19^11 × 23^9 × 29^7 × … (47 primes in all). The 53 trailing zeros come from pairing the 53 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 221 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 221! ÷ e, so about 36.8% of all 5.0473 × 10^423 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 221! yourself

In Excel or Google Sheets, type =FACT(221). 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(221) returns the exact integer with all 424 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.