365! Is a Big Number

For reasons I won’t get into, I was looking at the Birthday Problem today. The probability of two people out of a group of n people having the same birthday (month/day) is equal to

Birthday Problem Probability Equation

Notice that 365! item? And the subsequent (365-n)! denominator? That pesky “!” means “factorial” and you derive a factorial number by multiplying all of the integers from 1 to n together like this: 1 x 2 x 3 x … x n = n!

So:
1! = 1
2! = 1 x 2 = 2
3! = 1 x 2 x 3 = 6
4! = 1 x 2 x 3 x 4 = 24
10! = 1 x 2 x … x 10 = 3,628,800
20! = 2.43e18

Obviously these numbers get big fast. I’m familiar with the factorial so when I read “365!” I actually started laughing out loud. I don’t have access to any particular piece of equipment or software that is capable of calculating that number. Thankfully, Wikipedia has a nice table with a short breakdown of the results, depending on n. Now I know that if ten people are in a room, and assuming a normal distribution of birthdates (which is a faulty assumption), the probability of at least two people sharing a birthday is 11.7%. My day is complete.

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