Factorial of 12

Counting with 12!

There are 479,001,600 different ways to arrange 12 different books on a shelf: 12 choices for the first place, 11 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 12 runners, there are 12 × 11 × 10 = 1,320 possible podiums. If the order does not matter, divide by 3! = 6 to get 220 ways to choose a group of three. Choosing a pair from 12 people can be done in 66 ways.

In probability, if 12 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 479,001,600. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 12!?

12! has 9 digits, so it is in the millions to billions, around the number of people on Earth. The digits add up to 27, 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 12!

Every prime up to 12 divides 12!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 12! that gives 2^10 × 3^5 × 5^2 × 7 × 11. The 2 trailing zeros come from pairing the 2 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 12 items: orders in which no item ends up in its original place, as in a Secret Santa where nobody draws their own name. For 12 items there are exactly 176,214,841, which is 36.8% of all 479,001,600 orders. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 12! yourself

In Excel or Google Sheets, type =FACT(12). 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(12) returns the exact integer with all 9 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.