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Applications of Probability

Permutations and Combinations in Probability

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.

6 consonants and 5 vowels
He wants to create as many different arrangements as possible using 7 of the letters without repeating any letters in each arrangement. He also decides that the arrangements must consist of 3 vowels and 4 consonants. How many different arrangements can Vincenzo create?

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 4, 5, and 6, without repetitions. Any of the three digits can be picked for the first position, leaving 2 choices for the second position, and 1 choice for the third position.

example permutation
In this case, there are 6 possible permutations.
Although all these numbers are formed with the same three digits, the order in which they appear will make a different number. The number of permutations can be calculated by using the Fundamental Counting Principle.
Listing all the permutations may be a difficult task when having many objects. In these cases, the Permutation Formula can be used instead.

Rule

Permutation Formula

The number of permutations of n different objects arranged r 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 0!=1, when n=r the number of permutations is given by the factorial of n.

An alternative notation for is P(n,r).

Proof

Permutation Formula

The formula can be proven by using the Fundamental Counting Principle. In an arrangement with r elements, there are n choices for the first element, n1 choices for the second element, n2 choices for the third element, and so on.

Position Number of Choices
1 n
2 n1
3 n2
r (nr+1)
By the Fundamental Counting Principle, the product of the choices for each element is equal to the different arrangements of n objects chosen r at a time.
The right-hand side of this equation consists of the first r factors of the factorial of n. The Multiplication Property of Equality can be used to multiply both sides by the last nr factors of the factorial of n. The product of these factors can be written as (nr)!
Solve for

Write as a product

Write as a factorial

Example

Investigating Permutations in Real-Life Situations

The following cities are the ten most visited cities in Europe.

Rank City
1 London, UK
2 Paris, France
3 Istanbul, Turkey
4 Antalya, Turkey
5 Rome, Italy
6 Prague, Czech Republic
7 Amsterdam, Netherlands
8 Barcelona, Spain
9 Vienna, Austria
10 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?
b Suppose that they can only visit 3 of the ten places. In how many ways can they do it?

Hint

a The number of permutations of n out of n is given by the factorial of n.
b Consider the permutation formula for r objects out of n.

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 n items out of n is given by the factorial of n.
In this case, since there are ten cities to be visited, the factorial of 10 needs to be calculated.

Write as a product

Therefore, Dominika and Heichi have 3628800 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 r items out of n is given by the following formula.
Of the 10 places, only 3 can be visited. Therefore, the number of permutations of 3 out of 10 needs to be calculated.
Evaluate right-hand side

Write as a product

There are 720 ways of visit 3 of the ten cities.

Example

Finding Probabilities Using Permutations

In the 2020 Olympic Games, the competitors of the men's 100 meter freestyle swimming finals came from the following countries.

Men’s 100 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?
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.

Hint

a The order in which the medals are awarded is essential.
b The probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.

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 r items out of n is given by the following formula.
Of the 8 swimmers, only 3 can be on the podium and be awarded medals. Therefore, the number of permutations of 3 out of 8 needs to be calculated.
Evaluate right-hand side

Write as a product

8P3=876
8P3=336
There are 336 different ways in which the gold, silver, and bronze medals can be awarded.
b Recall that the probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes.
probability formula

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
If the medals are awarded to the Italian, French, and Australian swimmers in this order, then the number of permutations of the last five positions must be calculated.
The number of possible outcomes is the number of all possible ways in which the medals can be awarded. This is found by calculating the permutations of 8 swimmers taken from a group of 8.
With this information, the probability that the Italian swimmer wins the golden medal, the French swimmer the silver medal, and the Australian swimmer the bronze medal can be calculated.

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.

combinations salad
Because the order of the items does not matter, two combinations are different from each other if they do not have the same objects. The number of combinations can be found by listing every possible combination. However, this method is not helpful when considering a large number of objects. The Combination Formula should be used instead.

Rule

Combination Formula

The number of combinations of n different objects taken r 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 0!=1, when n=r the number of combinations is 1.

An alternative notation for is C(n,r).

Proof

The formula can be proven by using the Permutation Formulas.
Let be the number of combinations of n objects chosen r at a time. By the Fundamental Counting Principle, the product of by equals the number of permutations of r objects out of n.
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 4 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
In how many ways can they select 4 different books?

Hint

The order in which the books are selected is not crucial.

Solution

As long as 4 books are selected, the order is not important. Therefore, the different ways in which Kriz can select 4 books can be found by using combinations. The number of combinations when selecting r items out of n is given by the following formula.
By substituting 10 for n and 4 for r, the number of combinations can be calculated.
Evaluate right-hand side

Write as a product

Write as a product

There are 210 ways in which Kriz can select 4 books to pack from their must-read list.

Example

Finding Probabilities Using Combinations

Kriz has decided that they will select 5 of their books at random instead of 4. 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 10 books from 10 different genres? Write the answer in percentage form rounded to 1 decimal place.

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.

probability of an event formula
The order in which the books are selected is not important. Therefore, the number of possible outcomes can be found by calculating the combinations when taking 5 books out of 10. The number of combinations when selecting r items out of n is given by the following formula.
By substituting 10 for n and 5 for r, the number of possible outcomes can be calculated.
Evaluate right-hand side

Write as a product

Write as a product

There are 252 ways of selecting 5 books out of 10. This is the number of possible outcomes.
Since the order does not matter, there is only one way of selecting a fantasy, a mystery, and a drama book. The other 2 books need to be selected from the remaining 7. Therefore, the combinations of 2 out of 7 books will be calculated.
Evaluate right-hand side

Write as a product

2!=2

There is 1 way to select the first three books and 21 ways to select the other two books. By the Fundamental Counting Principle, the number of favorable outcomes is the product between 1 and 21.
There are 21 favorable outcomes and 252 possible outcomes. Let A be the event that 3 of the 5 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, P(A) can be found.
P(A)=0.083
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 10 students. While making a list to track their attendance, she wonders whether at least 2 students have the same birthday. For simplicity, suppose that all years have exactly 365 days.

list of students
a What is the probability that at least two students have the same birthday? Approximate the answer to two decimal places.
b If there were 30 students, what would be the probability that at least 2 students the same birthday? Approximate the answer to two decimal places.

Hint

b Substitute 30 for n and 2 for r in the expression found in Part A.

Solution

a Note that the phrase at least means that any outcome with 2 or more students with the same birthday is a favorable outcome. Therefore, the following outcomes are all the possible favorable outcomes.
Favorable Outcomes
2 have the same birthday 3 have the same birthday 4 have the same birthday
5 have the same birthday 6 have the same birthday 7 have the same birthday
8 have the same birthday 9 have the same birthday 10 have the same birthday
Suppose that instead of 10, there are n students. Let A be the event that at least 2 students out of n have their birthday on the same day. If n is greater than 365, then certainly at least 2 students would share their birthday. For the sake of the example, n is assumed to be less than or equal to 365.
Finding the probability of every possible outcome — all nine outcomes in the table above — means finding nine different probabilities. The opposite of at least 2 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 A 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 n having the same birthday. Then, P(A) can be calculated.

General Expression For and P(A)

Suppose that the group consists of only 2 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 364 to 365. 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 363 to 365. 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 n students do not share a birthday, can be written.
The obtained expression will now be simplified.
Simplify right-hand side
Now, the numerator will be simplified.
Simplify
The above expression represents the permutations of 365 taking n at a time.
Therefore, is the quotient of and
The Complement Rule of Probability states that P(A) is the difference between 1 and By using the expression found for an expression for P(A) can be found.

Calculating P(A) if n=10

In this case, A is the event that at least 2 students out of 10 have their birthday on the same day.

A
At leat 2 students out of 10 have their birthday on the same day. No one of the 10 students shares a birthday.
To apply the formula, will be calculated first.
Evaluate right-hand side

Write as a product

Finally, by subtracting from 1, the probability of A can be calculated.

b It was found that if A is the event that at least 2 out of n students have their birthday on the same day, P(A) is given by the following equation.
By substituting 30 for n, P(A) can be found for a group of 30 students.
Evaluate right-hand side

Write as a product

The probability that at least 2 students out of 30 have their birthday on the same day is about 0.71. What is more, and extremely curious, only 57 people are required to bring the probability that at least two people have the same birthday up to 0.99.

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.

6 consontants and 5 vowels
How many different arrangements with 3 vowels and 4 consonants can be created?

Hint

Begin by calculating the number of ways of selecting 3 vowels and 4 consonants. The order of the arrangements is essential.

Solution

Because the arrangements consist of 3 vowels and 4 consonants, they have 7 letters. An example arrangement is shown.

Example word with 3 vowels and 4 consonants from the ones given
Note that 3 vowels must be selected out of 5, which means that the number of possible combinations must be calculated. To do so, the combination formula can be used. The number of combinations of n objects taken r at a time is given by the following formula.
Therefore, the number of possible combinations can be calculated by substituting 5 for n and 3 for r into the formula.
Evaluate right-hand side

Write as a product

2!=2

In a similar way, the number of combinations when taking 4 out of 6 constants can be found.
Evaluate right-hand side

Write as a product

2!=2

There are 10 ways of selecting three vowels and 15 ways of selecting four consonants. By the Fundamental Counting Principle, the number of ways of selecting three vowels and four consonants is given by the product of 10 and 15.
Now, keep in mind that a different arrangement of each possible combination is different. This means that the order is important, and the number of possible permutations of the seven letters must to be calculated. Recall that the factorial of n gives the number of permutations of n out of n.
Finally the Fundamental Counting Principle will be used one more time. For each of the 150 ways of selecting three vowels and four consonants, there are 5040 permutations.
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