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# 2.3: Permutation Problems

Difficulty Level: Basic Created by: CK-12
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Practice Permutation Problems

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Suppose you want to select items to place on your shelf at home from left to right. If you have 45 items from which to select how many possible arrangements can you make? Can you see why a calculator might be useful to find this answer?

### Watch This

First watch this video to learn about calculating permutations with calculators.

Then watch this video to see some examples.

### Guidance

To calculate permutations (\begin{align*}nPr\end{align*}) on the TI calculator, first enter the \begin{align*}n\end{align*} value, and then press \begin{align*}\boxed{\text{MATH}}\end{align*}. You should see menus across the top of the screen. You want the fourth menu: PRB (arrow right 3 times). The PRB menu should appear as follows:

You will see several options, with \begin{align*}nPr\end{align*} being the second. Press \begin{align*}\boxed{2}\end{align*}, and then enter the \begin{align*}r\end{align*} value. Finally, press \begin{align*}\boxed{\text{ENTER}}\end{align*} to calculate the answer.

#### Example A

Compute \begin{align*}{_9}P_5\end{align*} using your TI calculator.

\begin{align*}\boxed{9} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{5} \ \boxed{\text{ENTER}}\end{align*}

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}, you should see the following on your calculator's screen:

Therefore, \begin{align*}{_9}P_5= 15,120\end{align*}.

#### Example B

In how many ways can first and second place be awarded to 10 people? Compute the answer using your TI calculator.

There are 10 people \begin{align*}(n = 10)\end{align*}, and there are 2 prize winners \begin{align*}(r = 2)\end{align*}, so to find the answer, enter the following into your TI calculator:

\begin{align*}\boxed{10} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{2} \ \boxed{\text{ENTER}}\end{align*}

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}, you should see the following on your calculator's screen:

Therefore, \begin{align*}{_{10}}P_2= 90\end{align*}, which means that the number of ways that first and second place can be awarded to 10 people is 90.

#### Example C

In how many ways can 3 favorite desserts be listed in order from a menu of 10? Compute the answer using your TI calculator.

There are 10 menu items \begin{align*}(n = 10)\end{align*}, and you are choosing 3 favorite desserts \begin{align*}(r = 3)\end{align*} in order, so to find the answer, enter the following into your TI calculator:

\begin{align*}\boxed{10} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{3} \ \boxed{\text{ENTER}}\end{align*}

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}, you should see the following on your calculator's screen:

Therefore, \begin{align*}{_{10}}P_3= 720\end{align*}, which means that the number of ways that 3 favorite desserts can be listed in order from a menu of 10 is 720.

### Guided Practice

Verify that the \begin{align*}nPr\end{align*} option on the PRB menu of the TI calculator works by calculating \begin{align*}{_{18}}P_{18}\end{align*} with both this option and with the factorial function. Note that with the TI calculator, you can find the factorial function by pressing \begin{align*}\boxed{\text{MATH}}\end{align*}, pressing the right arrow 3 times, and pressing \begin{align*}\boxed{4}\end{align*}.

First, lets compute \begin{align*}{_{18}}P_{18}\end{align*} with the factorial function. Remember, the formula to solve permutations like these is:

\begin{align*}{_n}P_r=\frac{n!}{(n-r)!}\end{align*}

This means that \begin{align*}{_{18}}P_{18}\end{align*} can be written as follows:

\begin{align*}{_{18}}P_{18} &= \frac{18!}{(18-18)!}\\ {_{18}}P_{18} &= \frac{18!}{0!}\\ {_{18}}P_{18} &= \frac{18!}{1}\\ {_{18}}P_{18} &= 18!\end{align*}

To find 18!, enter the following into your TI calculator:

\begin{align*}\boxed{1} \ \boxed{8} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{4} \ (\text{!}) \ \boxed{\text{ENTER}}\end{align*}

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}, you should see the following on your calculator's screen:

Now let's compute \begin{align*}{_{18}}P_{18}\end{align*} with the \begin{align*}nPr\end{align*} option on the PRB menu.

\begin{align*}\boxed{1} \ \boxed{8} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{1} \ \boxed{8} \ \boxed{\text{ENTER}}\end{align*}

After pressing \begin{align*}\boxed{\text{ENTER}}\end{align*}, you should see the following on your calculator's screen:

The answers for the 2 methods are the same, so the \begin{align*}nPr\end{align*} option on the PRB menu of the TI calculator does work.

### Practice

1. Enter each of the following sets of keystrokes into your TI calculator to compute the corresponding permutations.
2. \begin{align*}\boxed{1} \ \boxed{2} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{8} \ \boxed{\text{ENTER}}\end{align*}
3. \begin{align*}\boxed{1} \ \boxed{5} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{5} \ \boxed{\text{ENTER}}\end{align*}
4. \begin{align*}\boxed{2} \ \boxed{0} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{7} \ \boxed{\text{ENTER}}\end{align*}
5. \begin{align*}\boxed{1} \ \boxed{1} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{6} \ \boxed{\text{ENTER}}\end{align*}
6. \begin{align*}\boxed{1} \ \boxed{4} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{4} \ \boxed{\text{ENTER}}\end{align*}
7. \begin{align*}\boxed{1} \ \boxed{9} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{3} \ \boxed{\text{ENTER}}\end{align*}
8. \begin{align*}\boxed{2} \ \boxed{2} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{9} \ \boxed{\text{ENTER}}\end{align*}
9. \begin{align*}\boxed{1} \ \boxed{8} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{2} \ \boxed{\text{ENTER}}\end{align*}
10. \begin{align*}\boxed{2} \ \boxed{5} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{3} \ \boxed{\text{ENTER}}\end{align*}
11. \begin{align*}\boxed{1} \ \boxed{6} \ \boxed{\text{MATH}} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ \boxed{\blacktriangleright} \ (\text{PRB}) \ \boxed{2} \ (\text{nPr}) \ \boxed{6} \ \boxed{\text{ENTER}}\end{align*}

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### Vocabulary Language: English Spanish

TermDefinition
$n$ value When calculating permutations with the TI calculator, the $n$ value is the number of objects from which you are choosing, and the $r$ value is the number of objects chosen.
$r$ value When calculating permutations with the TI calculator, the $n$ value is the number of objects from which you are choosing, and the $r$ value is the number of objects chosen.
combination Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.
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 When calculating permutations with the TI calculator, the n value is the number of objects from which you are choosing.
Permutation A permutation is an arrangement of objects where order is important.
r value When calculating permutations with the TI calculator, the r value is the number of objects chosen.

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