Factorial of 9

Counting with 9!

There are 362,880 different ways to arrange 9 different books on a shelf: 9 choices for the first place, 8 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 9 runners, there are 9 × 8 × 7 = 504 possible podiums. If the order does not matter, divide by 3! = 6 to get 84 ways to choose a group of three. Choosing a pair from 9 people can be done in 36 ways.

In probability, if 9 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 362,880. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 9!?

9! has 6 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 9!

Every prime up to 9 divides 9!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 9! that gives 2^7 × 3^4 × 5 × 7. The 1 trailing zeros come from pairing the 1 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 9 items: orders in which no item ends up in its original place, as in a Secret Santa where nobody draws their own name. For 9 items there are exactly 133,496, which is 36.8% of all 362,880 orders. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 9! yourself

In Excel or Google Sheets, type =FACT(9). 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(9) returns the exact integer with all 6 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.