Factorial of 117

Counting with 117!

There are 3.9699 × 10^192 different ways to arrange 117 different books on a shelf: 117 choices for the first place, 116 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 117 runners, there are 117 × 116 × 115 = 1,560,780 possible podiums. If the order does not matter, divide by 3! = 6 to get 260,130 ways to choose a group of three. Choosing a pair from 117 people can be done in 6,786 ways.

In probability, if 117 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 3.9699 × 10^192. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 117!?

117! has 193 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 738, 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 117!

Every prime up to 117 divides 117!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 117! that gives 2^112 × 3^57 × 5^27 × 7^18 × 11^10 × 13^9 × 17^6 × 19^6 × 23^5 × 29^4 × … (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 117 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 117! ÷ e, so about 36.8% of all 3.9699 × 10^192 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 117! yourself

In Excel or Google Sheets, type =FACT(117). 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(117) returns the exact integer with all 193 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.