Factorial of 48

Counting with 48!

There are 1.2413 × 10^61 different ways to arrange 48 different books on a shelf: 48 choices for the first place, 47 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 48 runners, there are 48 × 47 × 46 = 103,776 possible podiums. If the order does not matter, divide by 3! = 6 to get 17,296 ways to choose a group of three. Choosing a pair from 48 people can be done in 1,128 ways.

In probability, if 48 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 1.2413 × 10^61. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 48!?

48! has 62 digits, so it is beyond everyday counting: it dwarfs the number of grains of sand on Earth (around 10^18). The digits add up to 234, 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 48!

Every prime up to 48 divides 48!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 48! that gives 2^46 × 3^22 × 5^10 × 7^6 × 11^4 × 13^3 × 17^2 × 19^2 × 23^2 × 29 × … (15 primes in all). The 10 trailing zeros come from pairing the 10 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 48 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 48! ÷ e, so about 36.8% of all 1.2413 × 10^61 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 48! yourself

In Excel or Google Sheets, type =FACT(48). 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(48) returns the exact integer with all 62 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.