Factorial of 51

Counting with 51!

There are 1.5511 × 10^66 different ways to arrange 51 different books on a shelf: 51 choices for the first place, 50 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 51 runners, there are 51 × 50 × 49 = 124,950 possible podiums. If the order does not matter, divide by 3! = 6 to get 20,825 ways to choose a group of three. Choosing a pair from 51 people can be done in 1,275 ways.

In probability, if 51 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 1.5511 × 10^66. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 51!?

51! has 67 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 198, 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 51!

Every prime up to 51 divides 51!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 51! that gives 2^47 × 3^23 × 5^12 × 7^8 × 11^4 × 13^3 × 17^3 × 19^2 × 23^2 × 29 × … (15 primes in all). The 12 trailing zeros come from pairing the 12 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 51 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 51! ÷ e, so about 36.8% of all 1.5511 × 10^66 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 51! yourself

In Excel or Google Sheets, type =FACT(51). 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(51) returns the exact integer with all 67 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.