Factorial of 350

Counting with 350!

There are 1.2358 × 10^740 different ways to arrange 350 different books on a shelf: 350 choices for the first place, 349 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 350 runners, there are 350 × 349 × 348 = 42,508,200 possible podiums. If the order does not matter, divide by 3! = 6 to get 7,084,700 ways to choose a group of three. Choosing a pair from 350 people can be done in 61,075 ways.

In probability, if 350 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 1.2358 × 10^740. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 350!?

350! has 741 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 2952, 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 350!

Every prime up to 350 divides 350!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 350! that gives 2^344 × 3^171 × 5^86 × 7^58 × 11^33 × 13^28 × 17^21 × 19^18 × 23^15 × 29^12 × … (70 primes in all). The 86 trailing zeros come from pairing the 86 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 350 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 350! ÷ e, so about 36.8% of all 1.2358 × 10^740 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 350! yourself

In Excel or Google Sheets, type =FACT(350). 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(350) returns the exact integer with all 741 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.