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### Combinations for Casting a Play

```
Date: 11/15/95 at 21:47:58
From: Anonymous
Subject: Permutations & Combinations

I gave this problem to a class of mine, but I now realize it is tad
more difficult than I first thought.  They however, are dying for an
answer.  Can you help?  I would like to show them the solution using
the fundamental counting principle.

The following shows the lists of people trying out for parts in
the school play. If someone does not get a part they may try out for
another if they have signed up. How many possible  casts are there?

Part  Ron     Alice    the butler   Grandpa Jones  Aunt Minnie
Jeeves
----------------------------------------------------------------
Billy   Susan      Sam        Ralph          Debbie
Sam     Sally      Ralph      Peter          Alison
Ralph   Alison     Peter      Joe            Tamara
Joe     Veronica   Bob                       Ruth
Debbie                               Lisa
```

```
Date: 11/19/95 at 14:44:28
From: Doctor Ken
Subject: Re: Permutations & Combinations

Hello!

Yes, you're right, this problem is kind of hairy.  Let me examine the
data:

There are three parts for the boys and two parts for the girls.  What we
can do is figure out how many ways there are to cast the boys' roles, and
how many ways there are to cast the girls, and then multiply these two
numbers together to find the total number of casts for the play.  So let's
look at the girls' roles first, since I bet they're easiest.

Alice     Minnie
-----------------
Susan     Lisa
Sally     Tamara
Veronica  Ruth
Alison    Alison
Debbie    Debbie

What we'll do is first treat the problem as if there were five people
trying out for Alice and five _different_ people trying out for Minnie;
then we'll get rid of all the combinations that had one person in two
different parts.

So let's do it: the number of pairs of roles without elimination is
5*5=25, and then there are two we'll have to eliminate: (Alison, Alison)
and (Debbie, Debbie).  So the total number of different combinations for
the girls' roles will be 25 - 2 = 23.

The same thinking will apply to the boys' roles, but it will be a little
more complicated.  Give it a try though, and see what happens.  If
you're still having trouble, write us back and let us know.  Good luck!

-Doctor Ken,  The Geometry Forum

```
Associated Topics:
High School Permutations and Combinations

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