Factorial of 128

Counting with 128!

There are 3.8562 × 10^215 different ways to arrange 128 different books on a shelf: 128 choices for the first place, 127 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 128 runners, there are 128 × 127 × 126 = 2,048,256 possible podiums. If the order does not matter, divide by 3! = 6 to get 341,376 ways to choose a group of three. Choosing a pair from 128 people can be done in 8,128 ways.

In probability, if 128 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 3.8562 × 10^215. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 128!?

128! has 216 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 828, 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 128!

Every prime up to 128 divides 128!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 128! that gives 2^127 × 3^61 × 5^31 × 7^20 × 11^12 × 13^9 × 17^7 × 19^6 × 23^5 × 29^4 × … (31 primes in all). The 31 trailing zeros come from pairing the 31 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 128 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 128! ÷ e, so about 36.8% of all 3.8562 × 10^215 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 128! yourself

In Excel or Google Sheets, type =FACT(128). 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(128) returns the exact integer with all 216 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.