There are N people. All with their own, unique hat. Each throw his hat into a pile, and mix it. The number of ways for n < N to get their hats is ( N-n)!. Why?

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- December 19th 2010, 08:56 PMphillipshwongProblem of hats
There are N people. All with their own, unique hat. Each throw his hat into a pile, and mix it. The number of ways for n < N to get their hats is ( N-n)!. Why?

- December 19th 2010, 09:07 PMpickslides
To understand these problems I take N to be a smallish number i.e. N=3, then find the number of arrangements for n=1,2.

Then take N=4 and find the number of arrangements for n=1,2,3

Finally take N=5 and find the number of arrangements for n=1,2,3,4 you'll start to see the pattern.

;D - December 20th 2010, 12:10 AMmr fantastic
- December 20th 2010, 06:10 AMSoroban
Hello, phillipshwong!

The problem is badly worded.

Perhaps it was written by an amateur, a student . . .

Quote:

II understand their reasoning . . .*think*

people get their hats back.

The other peopleget their hats back.*may or may not*

. . And there are ways to return those hats.

This is very sloppy work!

If they meantget their hats,*exactly*

. . the problem is more complicated.

The original problem did not consider this.

The other people mustget their own hats.*not*

. . This isof the hats,*derangement*

. . which can be written:

Example:

. . 8 people, 8 hats.

. . Find the number of ways that**exactly 3**get their hats.

There are: . choices of the 3 people.

The other 5 people mustget their own hats.*not*

. . There are: . ways. .**

Therefore, there are: . ways.

**

The Derangement Formula is a topic best left for a separate lesson.

Or you can do a search for "derangement".

- December 22nd 2010, 04:51 PMphillipshwong
- December 22nd 2010, 09:10 PMmr fantastic
- December 23rd 2010, 10:07 AMphillipshwong
That is ok, since I don 't have time for a course, or that concern with d(i). I already figure out why it is ( N-n)!