Factorial of 11

Counting with 11!

There are 39,916,800 different ways to arrange 11 different books on a shelf: 11 choices for the first place, 10 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 11 runners, there are 11 × 10 × 9 = 990 possible podiums. If the order does not matter, divide by 3! = 6 to get 165 ways to choose a group of three. Choosing a pair from 11 people can be done in 55 ways.

In probability, if 11 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 39,916,800. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 11!?

11! has 8 digits, so it is in the millions to billions, around the number of people on Earth. The digits add up to 36, 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 11!

Every prime up to 11 divides 11!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 11! that gives 2^8 × 3^4 × 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 11 items: orders in which no item ends up in its original place, as in a Secret Santa where nobody draws their own name. For 11 items there are exactly 14,684,570, which is 36.8% of all 39,916,800 orders. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 11! yourself

In Excel or Google Sheets, type =FACT(11). 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(11) returns the exact integer with all 8 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.