Factorial of 10

Counting with 10!

There are 3,628,800 different ways to arrange 10 different books on a shelf: 10 choices for the first place, 9 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 10 runners, there are 10 × 9 × 8 = 720 possible podiums. If the order does not matter, divide by 3! = 6 to get 120 ways to choose a group of three. Choosing a pair from 10 people can be done in 45 ways.

In probability, if 10 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 3,628,800. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 10!?

10! has 7 digits, so it is in the thousands to millions, about the size of a large city. 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 10!

Every prime up to 10 divides 10!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 10! that gives 2^8 × 3^4 × 5^2 × 7. 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 10 items: orders in which no item ends up in its original place, as in a Secret Santa where nobody draws their own name. For 10 items there are exactly 1,334,961, which is 36.8% of all 3,628,800 orders. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 10! yourself

In Excel or Google Sheets, type =FACT(10). 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(10) returns the exact integer with all 7 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.