Factorial of 135

Counting with 135!

There are 2.6904 × 10^230 different ways to arrange 135 different books on a shelf: 135 choices for the first place, 134 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 135 runners, there are 135 × 134 × 133 = 2,405,970 possible podiums. If the order does not matter, divide by 3! = 6 to get 400,995 ways to choose a group of three. Choosing a pair from 135 people can be done in 9,045 ways.

In probability, if 135 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 2.6904 × 10^230. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 135!?

135! has 231 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 819, 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 135!

Every prime up to 135 divides 135!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 135! that gives 2^131 × 3^66 × 5^33 × 7^21 × 11^13 × 13^10 × 17^7 × 19^7 × 23^5 × 29^4 × … (32 primes in all). The 33 trailing zeros come from pairing the 33 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 135 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 135! ÷ e, so about 36.8% of all 2.6904 × 10^230 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 135! yourself

In Excel or Google Sheets, type =FACT(135). 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(135) returns the exact integer with all 231 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.