7.8: Probability and Permutations
Suppose there are 15 people in a meeting, and one person will be the facilitator, while another person will be the scribe. In how many different ways can the facilitator and the scribe be chosen? What formula do you think you can use to answer this question? If the facilitator and the scribe were chosen at random, could you determine the probability of any particular pair of people? In this Concept, you'll learn to work with permutations and probability so that you can handle scenarios like this one.
Guidance
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(n1)(n2)\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*}
\begin{align*}_n P_k =\frac{n!}{(nk)!}\end{align*}
Example A
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 A can also be written using the formula for permutations: \begin{align*}_6P_3=\frac{6!}{(63)!}=\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.
Example B
Evaluate \begin{align*}_6P_3\end{align*}
Solution:
Type the first value of the permutation, the \begin{align*}n\end{align*}
Option #2 is the permutation option. Press [ENTER] and then the second value of the permutation, the value of \begin{align*}k\end{align*}
Permutations and Probability
Example C
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 Concept, 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*}
Video Review
Guided Practice
1. You have 8 different colors of nail polish. How many ways can you paint each of your 5 finger nails on one hand a different color?
2. Suppose 1 of your nail colors is purple. What is the probability that your thumb will be painted purple?
Solution:
1. There are 8 colors to paint, but you will only use 5 of them to paint your nails, which are in a particular order. This can be represented as \begin{align*}_8P_5\end{align*}
\begin{align*}_8P_5=8\cdot 7\cdot 6\cdot 5\cdot 4=6,720\end{align*}
2. The number of ways to paint your 5 nails different colors, where the thumb is purple, is calculated as:
\begin{align*}1\cdot 7\cdot 6\cdot 5\cdot \cdot 4=840 \end{align*}
Practice
 Define permutation.
In 2 – 19, evaluate each permutation.
 7!
 10!
 1!
 5!
 9!
 3!

\begin{align*}4!+4!\end{align*}
4!+4! 
\begin{align*}16!5!\end{align*}
16!−5! 
\begin{align*}\frac{98!}{96!}\end{align*}
98!96! 
\begin{align*}\frac{11!}{2!}\end{align*}
11!2! 
\begin{align*}\frac{301!}{300!}\end{align*}
301!300! 
\begin{align*}\frac{8!}{3!}\end{align*}
8!3! 
\begin{align*}2!+9!\end{align*}
2!+9! 
\begin{align*}_{11} P_2\end{align*}
11P2 
\begin{align*}_5P_5\end{align*}
5P5 
\begin{align*}_5P_3\end{align*}
5P3 
\begin{align*}_{15}P_{10}\end{align*}
15P10 
\begin{align*}_{60}P_{59}\end{align*}
60P59  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 can 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 threescoop 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 that you choose two cards (without replacement) from a standard 52card 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*}& 2x3y > 9\\
& y<1\end{align*}
2x−3y>−9y<1  Convert 24 meters/minute to feet/second.
 Solve for \begin{align*}t: t6 \le 14\end{align*}
t:t−6≤−14 .  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*}
12x+20y=cost: (3,6),(9,0),(6,2),(0,11) ?
Permutation
A permutation is an arrangement of objects where order is important.combination
Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.Favorable Outcome
A favorable outcome is the outcome that you are looking for in an experiment.Probability
Probability is the chance that something will happen. It can be written as a fraction, decimal or percent.Image Attributions
Here you'll learn how to find the number of permutations in a given situation and how to use this number in calculations of probability.
Concept Nodes:
Permutation
A permutation is an arrangement of objects where order is important.combination
Combinations are distinct arrangements of a specified number of objects without regard to order of selection from a specified set.Favorable Outcome
A favorable outcome is the outcome that you are looking for in an experiment.Probability
Probability is the chance that something will happen. It can be written as a fraction, decimal or percent.