Factorial of 115

Counting with 115!

There are 2.9250 × 10^188 different ways to arrange 115 different books on a shelf: 115 choices for the first place, 114 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 115 runners, there are 115 × 114 × 113 = 1,481,430 possible podiums. If the order does not matter, divide by 3! = 6 to get 246,905 ways to choose a group of three. Choosing a pair from 115 people can be done in 6,555 ways.

In probability, if 115 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 2.9250 × 10^188. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 115!?

115! has 189 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 648, 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 115!

Every prime up to 115 divides 115!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 115! that gives 2^110 × 3^55 × 5^27 × 7^18 × 11^10 × 13^8 × 17^6 × 19^6 × 23^5 × 29^3 × … (30 primes in all). The 27 trailing zeros come from pairing the 27 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 115 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 115! ÷ e, so about 36.8% of all 2.9250 × 10^188 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 115! yourself

In Excel or Google Sheets, type =FACT(115). 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(115) returns the exact integer with all 189 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.