Factorial of 3

Counting with 3!

There are 6 different ways to arrange 3 different books on a shelf: 3 choices for the first place, 2 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 3 runners, there are 3 × 2 × 1 = 6 possible podiums. If the order does not matter, divide by 3! = 6 to get 1 ways to choose a group of three. Choosing a pair from 3 people can be done in 3 ways.

In probability, if 3 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 6. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 3!?

3! has 1 digits, so it is a small number you can count to quickly. The digits add up to 6. 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 3!

Every prime up to 3 divides 3!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 3! that gives 2 × 3. The 0 trailing zeros come from pairing the 0 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 3 items: orders in which no item ends up in its original place, as in a Secret Santa where nobody draws their own name. For 3 items there are exactly 2, which is 33.3% of all 6 orders. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 3! yourself

In Excel or Google Sheets, type =FACT(3). 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(3) returns the exact integer with all 1 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.