#5**+10 **

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

heureka
Jan 8, 2016

#2**+10 **

**!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:

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

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

heureka
Jan 8, 2016

#3**0 **

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?

Melody
Jan 8, 2016

#5**+10 **

Best Answer

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

heureka
Jan 8, 2016

#6**0 **

Thanks very much Heureka,

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

--------------------

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 :)

Melody
Jan 8, 2016

#7**0 **

It's spelt "miss"

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

Coldplay
Jan 9, 2016

#8**+5 **

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

Dragonlance
Jan 9, 2016