Factorial of 188

Counting with 188!

There are 2.6957 × 10^347 different ways to arrange 188 different books on a shelf: 188 choices for the first place, 187 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 188 runners, there are 188 × 187 × 186 = 6,539,016 possible podiums. If the order does not matter, divide by 3! = 6 to get 1,089,836 ways to choose a group of three. Choosing a pair from 188 people can be done in 17,578 ways.

In probability, if 188 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 2.6957 × 10^347. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 188!?

188! has 348 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 1422, 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 188!

Every prime up to 188 divides 188!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 188! that gives 2^183 × 3^90 × 5^45 × 7^29 × 11^18 × 13^15 × 17^11 × 19^9 × 23^8 × 29^6 × … (42 primes in all). The 45 trailing zeros come from pairing the 45 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 188 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 188! ÷ e, so about 36.8% of all 2.6957 × 10^347 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 188! yourself

In Excel or Google Sheets, type =FACT(188). 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(188) returns the exact integer with all 348 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.