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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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Permutation Problems

### Permutation Problems

To calculate permutations (nPr\begin{align*}nPr\end{align*}) on the TI calculator, first enter the n\begin{align*}n\end{align*} value, and then press MATH\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 nPr\begin{align*}nPr\end{align*} being the second. Press 2\begin{align*}\boxed{2}\end{align*}, and then enter the r\begin{align*}r\end{align*} value. Finally, press ENTER\begin{align*}\boxed{\text{ENTER}}\end{align*} to calculate the answer.

#### Calculating Permutations

1. Compute 9P5\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*}.

2. 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.

3. 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.

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### Example

#### Example 1

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.

### Review

Enter each of the following sets of keystrokes into your TI calculator to compute the corresponding permutations.

1. \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*}
2. \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*}
3. \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*}
4. \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*}
5. \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*}
6. \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*}
7. \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*}
8. \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*}
9. \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*}
10. \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.