Factorial of 314

Counting with 314!

There are 2.0659 × 10^649 different ways to arrange 314 different books on a shelf: 314 choices for the first place, 313 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 314 runners, there are 314 × 313 × 312 = 30,663,984 possible podiums. If the order does not matter, divide by 3! = 6 to get 5,110,664 ways to choose a group of three. Choosing a pair from 314 people can be done in 49,141 ways.

In probability, if 314 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 2.0659 × 10^649. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 314!?

314! has 650 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 2601, 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 314!

Every prime up to 314 divides 314!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 314! that gives 2^309 × 3^153 × 5^76 × 7^50 × 11^30 × 13^25 × 17^19 × 19^16 × 23^13 × 29^10 × … (65 primes in all). The 76 trailing zeros come from pairing the 76 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 314 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 314! ÷ e, so about 36.8% of all 2.0659 × 10^649 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 314! yourself

In Excel or Google Sheets, type =FACT(314). 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(314) returns the exact integer with all 650 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.