Factorial of 49

Counting with 49!

There are 6.0828 × 10^62 different ways to arrange 49 different books on a shelf: 49 choices for the first place, 48 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 49 runners, there are 49 × 48 × 47 = 110,544 possible podiums. If the order does not matter, divide by 3! = 6 to get 18,424 ways to choose a group of three. Choosing a pair from 49 people can be done in 1,176 ways.

In probability, if 49 names are drawn out of a hat one at a time, the chance of drawing them in one particular order is 1 in 6.0828 × 10^62. That is why shuffled decks and passwords with many symbols are so hard to guess.

How big is 49!?

49! has 63 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 225, 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 49!

Every prime up to 49 divides 49!. Legendre's formula counts how often: add ⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + … for each prime p. For 49! that gives 2^46 × 3^22 × 5^10 × 7^8 × 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 49 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 49! ÷ e, so about 36.8% of all 6.0828 × 10^62 orders are derangements. From about four items on, that share stays close to 1 ÷ e, roughly 37%, however many items there are.

Computing 49! yourself

In Excel or Google Sheets, type =FACT(49). 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(49) returns the exact integer with all 63 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.