what is a derangement?

!!this is not a factorial

You have read that wrong, this is not 7 factorial but 7 bang!! Here I calculate the number of derangements of 7 objects, which is the number of ways of assigning 7 gifts to 7 people in such a way that each person doesn't receive their own gifts. In other words, it's the number of permutations of 7 objects without fixed points. I do it here by using a recursive argument, and in another video I prove this using the inclusion-exclusion principle. Enjoy this fun counting and probability and statistics extravaganza!!

[ianfowler]

Back when I was in University this was known as the "Montmort Letter Problem". The classical name for the derangement problem. Look it up. I was able to obtain the recursion relation in pretty much the same way you did. I was then able to develop the closed form or general nth term formula using Generating Functions and Infinite Power Series.

You can also derive this nth term formula, as you said, directly by using inclusion/exclusion - along with a little bit of inductive reasoning. This is how I did it on my first go. However you obtain the general nth term formula, it opens up a whole new way of understanding the problem. Very cool.

Anyway, thanks for re-introducing this famous classical problem of derangements to a younger and wider YouTube audience. You just gotta love this stuff.

[ClaraDeLemon]

Small detail you brushed over: in case 2, the second set of people and gifts is !(N-1) because people cannot be matched to their own gifts, except for P2, who could technically get P1's gift, but because case 2 specifically excludes this case, it can be treated as if it was their own gift they cannot give to themselves (as you said, you could relabel everything) and youd get the !(N-1) you were looking for. Great video as always!!

[Happy_Abe]

Loved this
It reminds me of the Airplane seats probability problem.

[vishalmishra]

*Factorial = Derangement x e hence directly proportional*
! N is the nearest integer close to N ! / e (and keeps getting closer as N increases to infinity or even greater than N=10).
You need to round up or down based on odd vs. even N.

[dougr.]

There MUST be an application to (or in) Group Theory!

[ShlokPatel_]

Spanish do call them factorials tho /s

Thank you for making fun videos πm <3