Factorial of 172

Counting with 172!

There are 2.1345 × 10^311 different ways to arrange 172 different books on a shelf: 172 choices for the first place, 171 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 172 runners, there are 172 × 171 × 170 = 5,000,040 possible podiums. If the order does not matter, divide by 3! = 6 to get 833,340 ways to choose a group of three. Choosing a pair from 172 people can be done in 14,706 ways.

In probability, if 172 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 2.1345 × 10^311. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 172!?

172! has 312 digits, so it is larger than the estimated number of atoms in the observable universe (about 10^80). The digits add up to 1215, 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 172!

Every prime up to 172 divides 172!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 172! that gives 2^168 × 3^84 × 5^41 × 7^27 × 11^16 × 13^14 × 17^10 × 19^9 × 23^7 × 29^5 × … (39 primes in all). The 41 trailing zeros come from pairing the 41 factors of 5 with factors of 2.

Shuffles where nothing stays put

A related count is the number of derangements of 172 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 172! ÷ e, so about 36.8% of all 2.1345 × 10^311 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 172! yourself

In Excel or Google Sheets, type =FACT(172). 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(172) returns the exact integer with all 312 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.