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### Permutations of Indistinguishable Objects

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Date: 09/21/2000 at 18:12:33
From: Candice
Subject: Permutations of Indistinguishable Objects and Multiplication
Principle

Hi Dr. Math,

I'm hoping you can help me out with two questions.

1. A forester selects four pink and four white dogwoods. The trees are
to be planted in row. If a tree is distinguished by color only, in how
many ways can the eight dogwoods be planted? How many of these
arrangements have at least two trees of the same color side by side?

I found the number of possibilities to be 576 (4x3x2x1x4x3x2x1), but
I don't know how to find the second answer.

2. A study is being designed to investigate the effect of polymer
ability to remove trace quantities of benzene from water. There are
2 polymer types (A and B), 3 temperatures (high, medium, low), 3
basic, neutral).

a) If each experimental condition is to replicated 5 times, how many

b) How many runs are made with polymer B at high or medium temperature
and low pH?

For a) I got 810 - number of possible conditions x 5. For b) I got
1x2x1x3x3 = 18. Is this correct?

```

```
Date: 09/21/2000 at 19:18:34
From: Doctor Anthony
Subject: Re: Permutations of Indistinguishable Objects and
Multiplication Principle

>If a tree is distinguished by color only, in how
>many ways can the eight dogwoods be planted?

The number of arrangements is  8!/[4!4!] = C(8,4) = 70

>How many of these
>arrangements have at least two trees of the same color side by side?

It is best to find the number of ways with no two of the same colour
side by side, and subtract this number from 70.

Suppose we place the 4 white trees in a row with spaces between them
and then place the pink trees as shown:

P *  P  *  P  *  P  *
or
*  P  *  P  *  P  *  P

and these are the only two ways that we have no two of the same colour
side by side. So the number of ways with at least two side by side is
68.

>2a) If each experimental condition is to replicated 5 times how many

5 x 2 x 3 x 3 x 3 x 3 = 810

>2b) How many runs are made with polymer B at high or medium
>temperature and low pH?

5 x 1 x 2 x 3 x 3 x 1 = 90

- Doctor Anthony, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Permutations and Combinations

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