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# Permutation Problems

## Using the nPr function found in the Math menu under PRB  on the TI calculator

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Practice Permutation Problems
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Evaluate Permutations Using Permutation Notation

Credit: Kenny Louie

Donna works at the library. She is arranging the new science fiction books that have come in. She only has room for six of the 20 new books on her shelf. How many different arrangements can be made when six books are chosen from a group of 20 books to be placed on a shelf?

In this concept, you will learn to evaluate permutations by using permutation notation.

### Guidance

The Counting Principle states that if event can be chosen in p\begin{align*}p\end{align*} different ways and an independent event is chosen in \begin{align*}q\end{align*} different ways, the probability of the two events occurring in \begin{align*}p \times q\end{align*}. Basically, this tells you how many ways you can arrange items. A permutation is a selection of items in which order is important. To use permutations to solve problems, you need to be able to identify the problems in which order, or the arrangement of items, matters.

Let’s look at an example.

How many ways can you arrange the letters in the word MATH?

First, look at the problem.

You have 4 letters in the word and you are going to choose one letter at a time. When you choose the first letter, you have 4 possibilities (M, A, T, or H), your second choice will be 3 possibilities, third choice will be 2 possibilities (because there are only 2 letters left to choose from), and last choice only one possibility.

Next, calculate the number of choices.

There are 24 different ways to arrange the letters in the word MATH.

The most efficient way to calculate permutations uses numbers called factorials. Factorials are special numbers that represent the product of a series of descending numbers.

The symbol for a factorial is an exclamation sign.

Take a look at how factorials are used.

\begin{align*}8 \ \text{factorial} = 8!= 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\end{align*}

\begin{align*}11 \ \text{factorial}= 11!= 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\end{align*}

\begin{align*}4 \ \text{factorial} = 4! = 4 \times 3 \times 2 \times 1\end{align*}

\begin{align*}17 \ \text{factorial} = 17! = 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \end{align*}

To compute the values of factorials, simply multiply the series of numbers.

You can also use a graphing calculator to find factorials. For large numbers, especially, the calculator can save you time.

If you push the \begin{align*}m\end{align*} button and on the top of the screen you will see PROB. #4 under the PROB menu is factorial.

Let’s try an example.

The answer is 3628800. Notice the key press history to tell you what was pressed to get 10!

You can use factorials to calculate permutations. Look at the formula for finding permutations.

In general, permutations are written as:

To compute \begin{align*}_nP_r\end{align*} you write:

Let’s look at an example.

Suppose you have 6 items and you want to know how many arrangements you can make with 4 of the items.

First, order matters in this problem, so you need to find the number of permutations there are in 6 items taken 4 at a time.

First, in permutation notation write the following.

\begin{align*}{\color{red}_6}P{\color{blue}_4} = \end{align*} 6 items taken 4 at a time

Next, calculate \begin{align*}_6P_4\end{align*}.

Notice that it is the product of the values in descending order that tells you how many permutations are possible. You can use this method to solve any number of permutations.

You can also use the graphing calculator to find \begin{align*}_nP_r\end{align*}. If you push the m button and on the top of the screen you will see PROB. #2 under the PROB menu is \begin{align*}_nP_r\end{align*}.

To find \begin{align*}_8P_3\end{align*} using the calculator, you would get the following. Notice the key press history to tell you what to press to do \begin{align*}_8P_3\end{align*}.

### Guided Practice

Find \begin{align*}_7P_3\end{align*}.

First, fill in the formula for \begin{align*}_nP_r\end{align*} where \begin{align*}n = 7\end{align*} and \begin{align*}r = 3\end{align*}

Next, compute \begin{align*}_7P_3\end{align*}.

### Examples

#### Example 1

Find \begin{align*}_4P_3\end{align*}.

First, fill in the formula for \begin{align*}_nP_r\end{align*} where \begin{align*}n = 4\end{align*} and \begin{align*}r = 3\end{align*}

Next, compute \begin{align*}_4P_3\end{align*}.

#### Example 2

Find \begin{align*}_{12}P_2\end{align*}.

First, fill in the formula for \begin{align*}_nP_r\end{align*} where \begin{align*}n = 12\end{align*} and \begin{align*}r = 2\end{align*}

Next, compute \begin{align*} _{12}P_2\end{align*}.

#### Example 3

Find \begin{align*}_8P_6\end{align*}.

First, fill in the formula for \begin{align*}_nP_r\end{align*} where \begin{align*}n = 8\end{align*} and \begin{align*}r = 6\end{align*}

Next, compute \begin{align*}_8P_6\end{align*}.

Credit: Andy Lamb

Remember Donna the librarian?

Donna is trying to choose six books of her 20 new books to put on the shelf. Remember that order is important and repetition is not allowed, so permutations must be used.

Donna has 20 books to choose from, therefore \begin{align*}n = 20\end{align*}. She will choose 6 books, therefore \begin{align*}r = 6\end{align*}.

First, fill in the formula for \begin{align*} _nP_r\end{align*} where \begin{align*}n = 20\end{align*} and \begin{align*}r = 6\end{align*}.

Next, compute \begin{align*}_{20}P_6\end{align*}.

There are 27,907,200 different ways Donna can choose six books of the 20 to display on the shelf.

### Explore More

Find each permutation.

1. Find \begin{align*}_7P_2\end{align*}

2. Find \begin{align*}_6P_3\end{align*}

3. Find \begin{align*}_5P_4\end{align*}

4. Find \begin{align*}_5P_5\end{align*}

5. Find \begin{align*}_9P_3\end{align*}

6. Find \begin{align*}_9P_7\end{align*}

7. Find \begin{align*}_{11}P_3\end{align*}

8. Find \begin{align*}_{12}P_3\end{align*}

9. Find \begin{align*}_6P_2\end{align*}

10. Find \begin{align*}_{14}P_3\end{align*}

11. Find \begin{align*}_{15}P_3\end{align*}

12. Find \begin{align*}_{11}P_4\end{align*}

13. Find \begin{align*}_{16}P_2\end{align*}

Use permutations to solve each problem.

14. Mia has 7 charms for her charm bracelet – a heart, a moon, a turtle, a cube, a bird, a hoop, and a car. Into how many different orders can she arrange the 7 charms?

15. One of the charms in Mia’s bracelet in problem 6 above fell off. How many fewer arrangements are there now?

### Vocabulary Language: English

combination

combination

Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.
factorial

factorial

The factorial of a whole number $n$ is the product of the positive integers from 1 to $n$. The symbol "!" denotes factorial. $n! = 1 \cdot 2 \cdot 3 \cdot 4...\cdot (n-1) \cdot n$.
n value

n value

When calculating permutations with the TI calculator, the n value is the number of objects from which you are choosing.
Permutation

Permutation

A permutation is an arrangement of objects where order is important.
r value

r value

When calculating permutations with the TI calculator, the r value is the number of objects chosen.