Factorial of 19

Counting with 19!

There are 121,645,100,408,832,000 different ways to arrange 19 different books on a shelf: 19 choices for the first place, 18 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 19 runners, there are 19 × 18 × 17 = 5,814 possible podiums. If the order does not matter, divide by 3! = 6 to get 969 ways to choose a group of three. Choosing a pair from 19 people can be done in 171 ways.

In probability, if 19 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 121,645,100,408,832,000. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 19!?

19! has 18 digits, so it is larger than the number of seconds since the universe began (about 4 × 10^17). The digits add up to 45, 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 19!

Every prime up to 19 divides 19!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 19! that gives 2^16 × 3^8 × 5^3 × 7^2 × 11 × 13 × 17 × 19. The 3 trailing zeros come from pairing the 3 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 19 items: orders in which no item ends up in its original place, as in a Secret Santa where nobody draws their own name. The count is the whole number nearest to 19! ÷ e, so about 36.8% of all 121,645,100,408,832,000 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 19! yourself

In Excel or Google Sheets, type =FACT(19). 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(19) returns the exact integer with all 18 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.