Factorial of 180

Counting with 180!

There are 2.0089 × 10^329 different ways to arrange 180 different books on a shelf: 180 choices for the first place, 179 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 180 runners, there are 180 × 179 × 178 = 5,735,160 possible podiums. If the order does not matter, divide by 3! = 6 to get 955,860 ways to choose a group of three. Choosing a pair from 180 people can be done in 16,110 ways.

In probability, if 180 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 2.0089 × 10^329. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 180!?

180! has 330 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 1305, 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 180!

Every prime up to 180 divides 180!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 180! that gives 2^176 × 3^88 × 5^44 × 7^28 × 11^17 × 13^14 × 17^10 × 19^9 × 23^7 × 29^6 × … (41 primes in all). The 44 trailing zeros come from pairing the 44 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 180 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 180! ÷ e, so about 36.8% of all 2.0089 × 10^329 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 180! yourself

In Excel or Google Sheets, type =FACT(180). 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(180) returns the exact integer with all 330 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.