!5 = ​

In combinatorial mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position.

The number of derangements of a set of size n, usually written Dn, dn, or !n, is called the "derangement number" or "de Montmort number". (These numbers are generalized to rencontres numbers.) The subfactorial function (not to be confused with the factorial n!) maps n to !n.

No standard notation for subfactorials is agreed upon; n¡ is sometimes used instead of !n.

see subfactorial or derangement  https://en.wikipedia.org/wiki/Derangement

see also: https://web2.0calc.com/questions/what-does-the-meaan

' Five Factorial'   is  5!  =  5 x 4 x 3 x 2 x 1 = 120

!5 = 44

\(\begin{array}{rcll} !n = n!\sum \limits_{k=0}^{n} \frac{(-1)^k}{k!} = n!\cdot \left( ~ 1 - \frac{1}{1!} +\frac{1}{2!} - \frac{1}{3!} + \cdots + (-1)^n \frac{1}{n!} ~ \right) \end{array}\)

Example:

\(\begin{array}{rcll} !6 = 6!\cdot \left( ~ 1 - \frac{1}{1!} +\frac{1}{2!} - \frac{1}{3!} +\frac{1}{4!} - \frac{1}{5!} +\frac{1}{6!}~ \right) = 265 \end{array}\)

\(\begin{array}{rcll} !5 &=& 5!\cdot \left( ~ 1 - \frac{1}{1!} +\frac{1}{2!} - \frac{1}{3!} +\frac{1}{4!} - \frac{1}{5!} ~ \right) \\ &=& 120\cdot \left( ~ 1 - 1 +\frac{1}{2} - \frac{1}{6} +\frac{1}{24} - \frac{1}{120} ~ \right) \\ &=& 120\cdot \left( ~ \frac{1}{2} - \frac{1}{6} +\frac{1}{24} - \frac{1}{120} ~ \right) \\ &=& 120\cdot \left( ~ \frac{3\cdot 4\cdot 5 - 4\cdot 5 + 5 - 1 }{120} ~ \right) \\ &=& 3\cdot 4\cdot 5 - 4\cdot 5 + 5 - 1 \\ &=& 60 - 20 + 5 - 1 \\ \mathbf{!5} & \mathbf{=} & \mathbf{44 } \end{array}\)

Thanks Heureka, I do not remember seeing that notation before.

Could someone please fill me in on what this notation is called and what main branch of mathematics would use it ://

I mean it would still be combinatory maths like n! is would it?

Hey Melody,

         Can't you google it?

Thanks very much Heureka,

So   !n   is called the derangement of n ?    :)

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Yes Mis Smartypants Coldplay I could just Google it.     X)

I google a great many things for people on here.

Sometimes it is easier, as well as nicer just to ask.

If i have a question then it is likely that other people have the same question.   :)

Once again, thanks very much Heureka :)

It's spelt "miss" 

And I am not a "smarty pants" I was just pointing out the obvious Ms. Melody!

Yep you are a smarty pants. The next step is smartbutt. That’s what I am and that is the nicer word for it. I not use the other word because Melody get really pissed when I use that kind of language on here. She even cusses at me for it. :)  I guess it is a type of derangement. hahaha

Remember to keep your smarty pants to cover your smartbutt. I still have mine but I too big for them. hahahaha

Now i have to deal with 2 of you      @%&@#!

I think that the feminine version is Misssssss Smartypants.  And the masculine version is Master Smart Bum!       Hahaha  !!