Factorial of 21

Counting with 21!

There are 51,090,942,171,709,440,000 different ways to arrange 21 different books on a shelf: 21 choices for the first place, 20 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 21 runners, there are 21 × 20 × 19 = 7,980 possible podiums. If the order does not matter, divide by 3! = 6 to get 1,330 ways to choose a group of three. Choosing a pair from 21 people can be done in 210 ways.

In probability, if 21 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 51,090,942,171,709,440,000. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 21!?

21! has 20 digits, so it is beyond everyday counting: it dwarfs the number of grains of sand on Earth (around 10^18). The digits add up to 63, 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 21!

Every prime up to 21 divides 21!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 21! that gives 2^18 × 3^9 × 5^4 × 7^3 × 11 × 13 × 17 × 19. The 4 trailing zeros come from pairing the 4 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 21 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 21! ÷ e, so about 36.8% of all 51,090,942,171,709,440,000 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 21! yourself

In Excel or Google Sheets, type =FACT(21). 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(21) returns the exact integer with all 20 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.