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In this lesson, the concepts of permutation and combination will be introduced and connected to the computation of probabilities of compound events.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
Challenge
Creating Arrangements
Vincenzo is playing with the following letters.
Discussion
Defining Permutations and the Permutation Formula
Many situations involve the rearrangement of a specific set of objects. These are called permutation problems. Below, the definition of permutation and its corresponding formula are discussed.
Concept
Permutation
A permutation is an arrangement of objects in which the order is important. For example, consider constructing a number using only the digits and without repetitions. Any of the three digits can be picked for the first position, leaving two choices for the second position, then only one choice for the third position.
Rule
Permutation Formula
The number of permutations of different objects arranged at a time — denoted as — is given by the following formula.
The exclamation sign in the formula indicates that the factorial of a value must be calculated. As a direct consequence, since when the number of permutations is given by the factorial of
An alternative notation for is
Proof
Permutation FormulaThe formula can be proven by using the Fundamental Counting Principle. In an arrangement with elements, there are choices for the first element, choices for the second element, choices for the third element, and so on.
| Position | Number of Choices |
|---|---|
Example
Investigating Permutations in Real-Life Situations
The following cities are the ten most visited cities in Europe.
| Rank | City |
|---|---|
| London, UK | |
| Paris, France | |
| Istanbul, Turkey | |
| Antalya, Turkey | |
| Rome, Italy | |
| Prague, Czech Republic | |
| Amsterdam, Netherlands | |
| Barcelona, Spain | |
| Vienna, Austria | |
| Milan, Italy |
a Dominika and her friend Heichi are planning to go to Europe next summer. How many different ways can they arrange their trip to see all ten cities?
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b Suppose that they can only visit of the ten places. In how many ways can they do it?
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Hint
a The number of permutations of out of is given by the factorial of
b Consider the permutation formula for objects out of
Solution
a Because the order in which the cities will be visited is important, the problem can be solved by using permutations. The number of permutations when taking items out of is given by the factorial of
In this case, since there are ten cities to be visited, the factorial of needs to be calculated.
Therefore, Dominika and Heichi have different ways to visit all ten most visited cities of Europe.
b Now suppose that Heichi and Dominika will visit only three places. Since the order of the cities they visit is important, permutations can be used again. The number of permutations when taking items out of is given by the following formula.
Of the places, only can be visited. Therefore, the number of permutations of out of needs to be calculated.
There are ways of visit of the ten cities.
▼
Evaluate right-hand side
SubTerm
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CancelCommonFac
Cancel out common factors
SimpQuot
Simplify quotient
Multiply
Multiply
Example
Finding Probabilities Using Permutations
In the Olympic Games, the competitors of the men's meter freestyle swimming finals came from the following countries.
| Men’s Meter Freestyle Swimming Finals | |
|---|---|
| United States | Australia |
| Russia | France |
| South Korea | Italy |
| Hungary | Romania |
a If there were no ties, in how many different ways could the gold, silver, and bronze medals have been awarded?
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b If all athletes have the same athletic ability, what is the probability that the Italian swimmer wins the gold medal, the French swimmer the silver medal, and the Australian swimmer the bronze medal? Approximate the answer to three decimals.
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Solution
a Because the order in which the medals are awarded is essential, the problem can be solved by using permutations. The number of permutations when taking items out of is given by the following formula.
▼
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b Recall that the probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.
The number of favorable outcomes is the number of ways the Italian, French, and Australian athletes can win the gold, silver, and bronze medals, respectively. Although there is only one way for the order of the first three positions, there are several ways for the order of the remaining five positions. All these are favorable outcomes.
| Example Favorable Outcomes | ||
|---|---|---|
| Italy | Italy | Italy |
| France | France | France |
| Australia | Australia | Australia |
| United States | Hungary | South Korea |
| Russia | Romania | Russia |
| South Korea | Russia | United States |
| Hungary | United States | Hungary |
| Romania | South Korea | Romania |
Discussion
Defining Combinations and the Combination Formula
In other situations, only the selected objects are important, not the order in which they come. These problems are called combination problems. Below, the definition of combination and its corresponding formula are developed.
Concept
Combination
A combination is a selection of objects in which the order is not important. Combinations focus on the selected objects. For example, consider choosing two different ingredients for a salad from five unique options in a salad bar.
Rule
Combination Formula
The number of combinations of different objects taken at a time — denoted as — is given by the following formula.
The exclamation mark in the formula indicates that the factorial of the value should be calculated. As a direct consequence of the above formula, since when the number of combinations is
An alternative notation for is
Proof
The formula can be proven by using the Permutation Formulas.
Let be the number of combinations of objects chosen at a time. By the Fundamental Counting Principle, the product of by equals the number of permutations of objects out of
Finally, by applying the Division Property of Equality, the Combination Formula is obtained.
Example
Investigating Combinations in Real-Life Situations
Kriz is going on vacation next month and wants to pack books from their must-read list. Each of the books belongs to one of the following genres.
| Kriz’s List of Books By Genres | |
|---|---|
| Fantasy | Romance |
| Mystery | Fiction |
| Biography | Graphic Novel |
| Drama | History |
| Western | Poetry |
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Hint
The order in which the books are selected is not crucial.
Solution
As long as books are selected, the order is not important. Therefore, the different ways in which Kriz can select books can be found by using combinations. The number of combinations when selecting items out of is given by the following formula.
By substituting for and for the number of combinations can be calculated.
There are ways in which Kriz can select books to pack from their must-read list.
▼
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SimpQuot
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Example
Finding Probabilities Using Combinations
Kriz has decided that they will select of their books at random instead of However, they would prefer to bring at least one book from the fantasy, mystery, and drama genres. What is the probability of them choosing these three genres if the selection pool consists of books from different genres? Write the answer in percentage form rounded to decimal place.
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Hint
The probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.
Solution
The probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.
▼
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of the books selected are a fantasy, a mystery, and a drama.By substituting the number of favorable outcomes and the number of possible outcomes into the Probability Formula, can be found. The probability of getting a fantasy, a mystery, and a drama in the five selected books is about
Example
The Birthday Problem
Magdalena teaches algebra to a group of students. While making a list to track their attendance, she wonders whether at least students have the same birthday. For simplicity, suppose that all years have exactly days.
a What is the probability that at least two students have the same birthday? Approximate the answer to two decimal places.
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b If there were students, what would be the probability that at least students the same birthday? Approximate the answer to two decimal places.
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Hint
a Consider the Complement Rule of Probability.
b Substitute for and for in the expression found in Part A.
Solution
a Note that the phrase
Suppose that instead of there are students. Let be the event that at least students out of have their birthday on the same day. If is greater than then certainly at least students would share their birthday. For the sake of the example, is assumed to be less than or equal to
Finding the probability of every possible outcome — all nine outcomes in the table above — means finding nine different probabilities. The opposite of at least students having their birthday on the same day is that there are no students with their birthday on the same day. This is the complement of written as
The Complement Rule of Probability can help to deal with this situation. To do so, a general expression will be found for no students from a group of having the same birthday. Then, can be calculated.
The obtained expression will now be simplified.
Now, the numerator will be simplified.
The above expression represents the permutations of taking at a time.
Therefore, is the quotient of and
The Complement Rule of Probability states that is the difference between and By using the expression found for an expression for can be found.
To apply the formula, will be calculated first.
Finally, by subtracting from the probability of can be calculated.
at leastmeans that any outcome with or more students with the same birthday is a favorable outcome. Therefore, the following outcomes are all the possible favorable outcomes.
| Favorable Outcomes | ||
|---|---|---|
| have the same birthday | have the same birthday | have the same birthday |
| have the same birthday | have the same birthday | have the same birthday |
| have the same birthday | have the same birthday | have the same birthday |
General Expression For and
Suppose that the group consists of only students. The first student can have their birthday on any day. The probability of the second student not having their birthday on the same day is the ratio of to Therefore, this expression is the probability of the two students not having their birthday on the same day. Suppose that a third student is added to the group. The probability that this student has their birthday on a different day from the previous two is the ratio of to By the Multiplication Rule of Probability, the probability that no students out of these three have their birthdays on the same day can be found. By following the same reasoning, the probability that students do not share a birthday, can be written.▼
Simplify right-hand side
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Simplify
Calculating if
In this case, is the event that at least students out of have their birthday on the same day.
| At leat students out of have their birthday on the same day. | No one of the students shares a birthday. |
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b It was found that if is the event that at least out of students have their birthday on the same day, is given by the following equation.
By substituting for can be found for a group of students.
The probability that at least students out of have their birthday on the same day is about What is more, and extremely curious, only people are required to bring the probability that at least two people have the same birthday up to
Closure
Creating Arrangements
Permutations and combinations can be used in many situations. Understanding these mathematical concepts can help solve many intricate problems. With this in mind, reconsider the problem in which Vincenzo wants to create arrangements by using the following letters.
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Hint
Begin by calculating the number of ways of selecting vowels and consonants. The order of the arrangements is essential.
Solution
Because the arrangements consist of vowels and consonants, they have letters. An example arrangement is shown.
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