Factorial of 210

Counting with 210!

There are 1.0582 × 10^398 different ways to arrange 210 different books on a shelf: 210 choices for the first place, 209 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 210 runners, there are 210 × 209 × 208 = 9,129,120 possible podiums. If the order does not matter, divide by 3! = 6 to get 1,521,520 ways to choose a group of three. Choosing a pair from 210 people can be done in 21,945 ways.

In probability, if 210 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 1.0582 × 10^398. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 210!?

210! has 399 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 1548, 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 210!

Every prime up to 210 divides 210!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 210! that gives 2^206 × 3^102 × 5^51 × 7^34 × 11^20 × 13^17 × 17^12 × 19^11 × 23^9 × 29^7 × … (46 primes in all). The 51 trailing zeros come from pairing the 51 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 210 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 210! ÷ e, so about 36.8% of all 1.0582 × 10^398 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 210! yourself

In Excel or Google Sheets, type =FACT(210). 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(210) returns the exact integer with all 399 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.