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What does ! mean in maths?

December 27, 2011

I’m convinced that by using Twitter for the last year (ish) I’ve been prompted to find out more things about maths than I have in many years without using Twitter.

So, the ! sign in maths has 2 uses. That’s right for all you smug people (like me) that were thinking “That’s easy, it’s Factorial” – two uses.

The Factorial of a natural number is what you get when you multiply that number by each one before it until you get to 1. For example:
3! = 3 x 2 x 1 = 6
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720
1! = 1
And through mathematical logic, 0! is defined to be equal to 1 (even though that makes no sense given the way I described factorial above).

The second use of ! is called the Subfactorial of a number and the sign comes before the (natural) number. The Subfactorial of a number is the permutations of that number of objects that leaves none of them in their original places. It would seem that these sorts or permutations are also called Derangements (presumably because none of the objects are arranged at all).
For example, if we consider two objects, then the permutations are (1,2) and (2,1). Clearly, the first one has objects in their correct place and the second one has none in their correct places. So:
!2 = 1

For 3 objects:
(1,2,3) (1,3,2) (2,1,3) (2,3,1) (3,1,2) (3,2,1)
Out of the 6 permutations, only (2,3,1) and (3,1,2) leave no objects in their correct places. So:
!3 = 2

In fact:
!1 = 0 !2 = 1
!3 = 2
!4 = 9 (you might to check this yourself.)

I’m not exactly sure where subfactorials are used but it would certainly seem that (theoretical) questions like the following would apply:

“A secretary has 5 letters and 5 envelopes and very little time. The letters get randomly put into an envelope. What is the probability that no letters end up in the correct envelope?”
Well, if my understanding of factorial and subfactorials are correct, then the answer to this question is the rather cool-looking:


We are all still learning maths and it wouldn’t hurt to let your students see you learning something new. I’ll certainly be sharing this as ‘something I did over the holidays’.


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  1. Shawn Urban permalink

    Hi Dave,Subfactorials are new to me too. And to think they have been around since the 1700’s. They seem rather obscure as there remain two operation signs for them – !n and n(upsidedown !) – even after 300 years. But I am assured they are used in permutations and associated probability, combinatorics and problems such as the one you asked above. (By the way, I think the answer to that one, according to Wikipedia ( is !5 = 44.) !n/n! limits to 1/e and, of course, n!/!n limits to e. !n and n! are also congruent modulo n-1. This after one hour of research. I am still looking for that magical WOW of subfactorials though. Always love discovering something new.Shawn

  2. chris_1974 permalink

    Amazing. I’ve heard of subfactorials somewhere along the line, but never knowingly used them, or seen the notation, or even understood what they are.I’ve come across the letter problem before, and am going to add it to my list of christmas (elves letters to santa etc…) lessons for next year. I’m also certainly going to mention subfactorials in passing to my y12 when we do S1 / S2. Just for fun!I agree, we need to share the delight of learning something new….

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