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Date: 02/25/2002 at 23:47:53
From: Bryan Smoot
Subject: Statistics probability

This is two-part question is confusing me.

There are 20 singers trying out for a musical. In how many different
ways can the director choose a duet?

In how many ways can the director choose a lead singer and a backup?

I thought the answers would be the same for both parts because it
would always end up being two people chosen, but apparently this is

Thanks.
```

```
Date: 02/27/2002 at 19:38:56
From: Doctor Tim
Subject: Re: Statistics probability

Hi!

This is a great question because it illuminates a really touchy issue
in this kind of problem - I'm going to have to remember it for my
students.

Let's try a simpler case:

Consider if there are only two singers, A and B. How many ways can
there be a duet? Only one. But there are two ways (AB, BA) to choose a
lead and backup. That is, in the second situation, ORDER MATTERS.

For a moment, let's do 3 singers.
Duet: AB AC BC. But order doesn't matter. 3 ways.
Lead/backup: AB BA AC CA BC CB. Order does matter. 6 ways.

Will it always be twice as many ways to do lead/backup? Sure. Because
every duet has two lead/backup configurations, and none of them
duplicates the lead/backup configurations for any other duet.

SO: how many ways can you choose 2 people out of 20? Let's do lead/
backup FIRST:

20 ways to pick the lead.
For each way, there are 19 ways to pick the backup.
So the answer is 20 * 19 or 380.

Now, from what we did above, we know that the number of duets is half
that: 190.

NOTE that this way of counting works fine, but it is also the same as
the combinations formula you probably know (which is for order-
doesn't-matter, the duet situation):

C(n, k) = n! / [(n-k)! k!]

C(20,2) = 20! /[(20-2)! 2!] = 20!/(18! 2!) = 20 * 19 / 2 = 190.

It took me YEARS to realize that the whole reason for the (n-k) in the
denominator was to cancel all but the first few numbers in the n!.
That is, you don't really want 20!, you just want 20*19. So one way to
get that is to call it 20!/18! - which cancels all the factors from 18
down.

I hope this helps!

- Doctor Tim, The Math Forum
http://mathforum.org/dr.math/
```

```
Date: 03/01/2002 at 12:10:12
From: Bryan Smoot
Subject: Statistics probability

provided much clarity to this confusing problem. Keep up the good
work!
```
Associated Topics:
High School Discrete Mathematics
High School Permutations and Combinations

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