Factorial of 84

Counting with 84!

There are 3.3142 × 10^126 different ways to arrange 84 different books on a shelf: 84 choices for the first place, 83 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 84 runners, there are 84 × 83 × 82 = 571,704 possible podiums. If the order does not matter, divide by 3! = 6 to get 95,284 ways to choose a group of three. Choosing a pair from 84 people can be done in 3,486 ways.

In probability, if 84 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 3.3142 × 10^126. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 84!?

84! has 127 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 477, 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 84!

Every prime up to 84 divides 84!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 84! that gives 2^81 × 3^41 × 5^19 × 7^13 × 11^7 × 13^6 × 17^4 × 19^4 × 23^3 × 29^2 × … (23 primes in all). The 19 trailing zeros come from pairing the 19 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 84 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 84! ÷ e, so about 36.8% of all 3.3142 × 10^126 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 84! yourself

In Excel or Google Sheets, type =FACT(84). 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(84) returns the exact integer with all 127 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.