Factorial of 265

Counting with 265!

There are 4.8191 × 10^528 different ways to arrange 265 different books on a shelf: 265 choices for the first place, 264 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 265 runners, there are 265 × 264 × 263 = 18,399,480 possible podiums. If the order does not matter, divide by 3! = 6 to get 3,066,580 ways to choose a group of three. Choosing a pair from 265 people can be done in 34,980 ways.

In probability, if 265 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 4.8191 × 10^528. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 265!?

265! has 529 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 2106, 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 265!

Every prime up to 265 divides 265!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 265! that gives 2^262 × 3^130 × 5^65 × 7^42 × 11^26 × 13^21 × 17^15 × 19^13 × 23^11 × 29^9 × … (56 primes in all). The 65 trailing zeros come from pairing the 65 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 265 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 265! ÷ e, so about 36.8% of all 4.8191 × 10^528 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 265! yourself

In Excel or Google Sheets, type =FACT(265). 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(265) returns the exact integer with all 529 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.