# 7.7: Probability and Permutations

**At Grade**Created by: CK-12

Congratulations! You have won a free trip to Europe. On your trip you have the opportunity to visit 6 different cities. You are responsible for planning your European vacation. How many different ways can you schedule your trip? The answer may surprise you!

This is an example of a **permutation**.

A **permutation** is an arrangement of objects in a specific order. It is the product of the counting numbers 1 through \begin{align*}n\end{align*}

\begin{align*}n!=n(n-1)(n-2)\cdot \ldots \cdot 1\end{align*}

How many ways can you visit the European cities? There are 6 choices for the first stop. Once you have visited this city, you cannot return so there are 5 choices for the second stop, and so on.

\begin{align*}\underline{6} \cdot \underline{5} \cdot \underline{4} \cdot \underline{3} \cdot \underline{2} \cdot \underline{1}=720\end{align*}

There are 720 different ways to plan your European vacation!

A permutation of \begin{align*}n\end{align*}** objects** arranged \begin{align*}k\end{align*}

**is expressed as \begin{align*}_n P_k\end{align*}**

*at a time*
\begin{align*}_n P_k =\frac{n!}{(n-k)!}\end{align*}

**Example 1:** *Evaluate* \begin{align*}_6P_3\end{align*}

**Solution:** This equation asks, “How many ways can 6 objects be chosen 3 at a time?”

There are 6 ways to choose the first object, 5 ways to choose the second object, and 4 ways to choose the third object.

\begin{align*}\underline{6} \cdot \underline{5} \cdot \underline{4}=120\end{align*}

There are 120 different ways 6 objects can be chosen 3 at a time.

Example 1 can also be written using the formula for permutation: \begin{align*}_6P_3=\frac{6!}{(6-3)!}=\frac{6!}{3!}=6 \cdot 5 \cdot 4=120\end{align*}

## Permutations and Graphing Calculators

Most graphing calculators have the ability to calculate a permutation.

*Evaluate \begin{align*}_6P_3\end{align*} using a graphing calculator.*

Type the first value of the permutation, the \begin{align*}n\end{align*}. Choose the **[MATH]** button, directly below the **[ALPHA]** key. Move the cursor once to the left to see this screen:

Option #2 is the permutation option. Press **[ENTER]** and then the second value of the permutation, the value of \begin{align*}k\end{align*}. Press **[ENTER]** to evaluate.

## Permutations and Probability

The letters of the word HOSPITAL are arranged at random. How many different arrangements can be made? What is the probability that the last letter is a vowel?

There are eight ways to choose the first letter, seven ways to choose the second, and so on. The total number of arrangements is 8!= 40,320.

There are three vowels in HOSPITAL; therefore, there are three possibilities for the last letter. Once this letter is chosen, there are seven choices for the first letter, six for the second, and so on.

\begin{align*}\underline{7} \cdot \underline{6} \cdot \underline{5} \cdot \underline{4} \cdot \underline{3} \cdot \underline{2} \cdot \underline{1} \cdot \underline{3}=15,120\end{align*}

Probability, as you learned in a previous chapter, has the formula:

\begin{align*}\text{Probability} \ (success) = \frac{number \ of \ ways \ to \ get \ success}{total \ number \ of \ possible \ outcomes}\end{align*}

There are 15,120 ways to get a vowel as the last letter; there are 40,320 total combinations.

\begin{align*}P(last \ letter \ is \ a \ vowel)=\frac{15,120}{40,320}=\frac{3}{8}\end{align*}

**Multimedia Link:** For more help with permutations, visit the http://regentsprep.org/REgents/math/ALGEBRA/APR2/LpermProb.htm - Algebra Lesson Page by Regents Prep.

## Practice Set

- Define
*permutation*.

In 2 – 19, evaluate each permutation.

- 7!
- 10!
- 1!
- 5!
- 9!
- 3!
- \begin{align*}4!+4!\end{align*}
- \begin{align*}16!-5!\end{align*}
- \begin{align*}\frac{98!}{96!}\end{align*}
- \begin{align*}\frac{11!}{2!}\end{align*}
- \begin{align*}\frac{301!}{300!}\end{align*}
- \begin{align*}\frac{8!}{3!}\end{align*}
- \begin{align*}2!+9!\end{align*}
- \begin{align*}_{11} P_2\end{align*}
- \begin{align*}_5P_5\end{align*}
- \begin{align*}_5P_3\end{align*}
- \begin{align*}_{15}P_{10}\end{align*}
- \begin{align*}_{60}P_{59}\end{align*}
- How many ways can 14 books be organized on a shelf?
- How many ways are there to choose 10 objects, four at a time?
- How many ways are there to choose 21 objects, 13 at a time?
- A running track has eight lanes. In how many ways can 8 athletes be arranged to start a race?
- Twelve horses run a race.
- How many ways can first and second places be won?
- How many ways will all the horses finish the race?

- Six actors are waiting to audition. How many ways can the director choose the audition schedule?
- Jerry, Kerry, Larry, and Mary are waiting at a bus stop. What is the probability that Mary will get on the bus first?
- How many permutations are there of the letters in the word “HEART”?
- How many permutations are there of the letters in the word “AMAZING”?
- Suppose I am planning to get a three-scoop ice cream cone with chocolate, vanilla, and Superman. How many ice cream cones are possible? If I ask the server to “surprise me,” what is the probability that the Superman scoop will be on top?
- What is the probability you choose two cards (without replacement) from a standard 52-card deck and both cards are jacks?
- The Super Bowl Committee has applications from 9 towns to host the next two Super Bowls. How many ways can they select the host if:
- The town cannot host a Super Bowl two consecutive years?
- The town can host a Super Bowl two consecutive years?

**Mixed Review**

- Graph the solution to the following system: \begin{align*}& 2x-3y > -9\\ & y<1\end{align*}
- Convert 24
*meters/minute*to*feet/second*. - Solve for \begin{align*}t: |t-6| \le -14\end{align*}.
- Find the distance between 6.15 and –9.86.
- Which of the following vertices provides the minimum cost according to the equation \begin{align*}12x+20y=cost: \ (3,6),(9,0),(6,2),(0,11)\end{align*}?
- Write the system of inequalities pictured below.

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