Factorial of 415

Counting with 415!

There are 9.2459 × 10^907 different ways to arrange 415 different books on a shelf: 415 choices for the first place, 414 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 415 runners, there are 415 × 414 × 413 = 70,957,530 possible podiums. If the order does not matter, divide by 3! = 6 to get 11,826,255 ways to choose a group of three. Choosing a pair from 415 people can be done in 85,905 ways.

In probability, if 415 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 9.2459 × 10^907. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 415!?

415! has 908 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 3708, 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 415!

Every prime up to 415 divides 415!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 415! that gives 2^408 × 3^205 × 5^102 × 7^68 × 11^40 × 13^33 × 17^25 × 19^22 × 23^18 × 29^14 × … (80 primes in all). The 102 trailing zeros come from pairing the 102 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 415 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 415! ÷ e, so about 36.8% of all 9.2459 × 10^907 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 415! yourself

In Excel or Google Sheets, type =FACT(415). 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(415) returns the exact integer with all 908 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.