Factorial of 457

Counting with 457!

There are 6.8905 × 10^1018 different ways to arrange 457 different books on a shelf: 457 choices for the first place, 456 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 457 runners, there are 457 × 456 × 455 = 94,818,360 possible podiums. If the order does not matter, divide by 3! = 6 to get 15,803,060 ways to choose a group of three. Choosing a pair from 457 people can be done in 104,196 ways.

In probability, if 457 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 6.8905 × 10^1018. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 457!?

457! has 1019 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 4176, 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 457!

Every prime up to 457 divides 457!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 457! that gives 2^452 × 3^224 × 5^112 × 7^75 × 11^44 × 13^37 × 17^27 × 19^25 × 23^19 × 29^15 × … (88 primes in all). The 112 trailing zeros come from pairing the 112 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 457 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 457! ÷ e, so about 36.8% of all 6.8905 × 10^1018 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 457! yourself

In Excel or Google Sheets, type =FACT(457). 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(457) returns the exact integer with all 1019 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.