Factorial of 123

Counting with 123!

There are 1.2146 × 10^205 different ways to arrange 123 different books on a shelf: 123 choices for the first place, 122 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 123 runners, there are 123 × 122 × 121 = 1,815,726 possible podiums. If the order does not matter, divide by 3! = 6 to get 302,621 ways to choose a group of three. Choosing a pair from 123 people can be done in 7,503 ways.

In probability, if 123 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 1.2146 × 10^205. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 123!?

123! has 206 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 765, 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 123!

Every prime up to 123 divides 123!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 123! that gives 2^117 × 3^59 × 5^28 × 7^19 × 11^12 × 13^9 × 17^7 × 19^6 × 23^5 × 29^4 × … (30 primes in all). The 28 trailing zeros come from pairing the 28 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 123 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 123! ÷ e, so about 36.8% of all 1.2146 × 10^205 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 123! yourself

In Excel or Google Sheets, type =FACT(123). 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(123) returns the exact integer with all 206 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.