Fundamental counting principle exercises

Study with the list of exercises on the fundamental principle of counting with jig.

The fundamental principle of counting is a mathematical tool in the area of ​​combinatorics. To understand and do well on assessments, it is important to practice. Enjoy and clear your doubts with the commented answers.

question 1

A pizzeria offers the following options of pizza flavors: chicken, pepperoni, ham and vegetarian. In addition, the pizzeria offers three sizes of pizza: small, medium and large. How many different pizza compositions can we create?

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Answer: 12 compositions.

For each flavor there are three size options. We can use the fundamental counting principle to solve the problem.

We have two independent choices: the flavor choice, with four possibilities, and the size choice, with three options.

So the total number of possible pizza combinations is:

4 (flavor options) x 3 (size options) = 12

So there are 12 different pizza combinations that can be made in the pizzeria.

question 2

Consider that a person has 3 shirts of different colors (red, blue and white), 2 pants of different models (jeans and dress) and 2 shoes of different types (sneakers and dress shoes). In how many different ways can this person dress?

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Answer: 12 combinations

The choices of shirt, pants and shoes are independent. This means that the choice of shirt color is not a limiting factor for the choice of pants and shoes.

Applying the fundamental counting principle, we have

3 shirts x 2 pants x 2 shoes = 12 combinations

question 3

A candy shop offers 4 flavors of ice cream (chocolate, strawberry, vanilla, and cream) and 3 toppings (chocolate sauce, caramel sauce, and whipped cream). How many different combinations of ice cream and frosting can you make in the store?

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Answer: 12 combinations.

4 (ice cream options) x 3 (topping options) = 12

So there are 12 different frosting ice cream combinations that can be made in the store.

question 4

A student needs to choose two extracurricular activities to participate in school, one cultural and one sporting. He can choose between the Theater Club, the Music Club or the Dance Club. In addition, he must choose either the Football team or the Volleyball team. How many different choices can the student make?

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Answer: 6 different choices.

3 cultural activities x 2 sports activities = 6

question 5

A person will travel by plane between two cities where it is necessary to make a connection, as no company offers direct flights. From city A to city B, where the connection will be made, three airlines offer flight options. From city B to C, four other companies make this journey.

How many different ways can this passenger travel from A to C and back to A using different flights?

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Answer: 72 options.

From A to B there are 3 options and from B to C there are 4 options. By the fundamental principle of counting, the forward path has:

3. 4 = 12 options

To return from C to B, without repeating the same flight, there are three options, because of the four that connected these two cities, one has already been used.

From city B to A there are 2 options that haven't been used yet. For the back there are:

3. 2 = 6 options

In total there will be:

12. 6 = 72 options

question 6

(Enem 2022) A car manufacturer disclosed that it offers its customers more than 1,000 different car configurations, varying the model, engine, options and color of the vehicle. Currently, it offers 7 car models with 2 types of engines: 1.0 and 1.6. Regarding options, there are 3 possible choices: multimedia center, alloy wheels and leather seats, the customer can choose to include one, two, three or none of the options available.

To be faithful to the announcement made, the minimum number of colors that the assembler must make available to its customers is

a) 8.

b) 9.

11.

18.

24.

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Answer explained

There are 7 model options and 2 engines.

Regarding the options: leather seats, alloy wheels and multimedia center, it is possible to choose three, two, one and none.

• Leather seats, alloy wheels and multimedia center;
• Leather seats and multimedia center;
• Leather seats and alloy wheels;
• Alloy wheels and multimedia center;
• leather seats;
• alloy wheels;
• Multimedia center;
• None.

Thus, regarding the options, there are 8 possible choices.

Applying the fundamental principle of counting and taking the number of colors as x, we have:

So there should be 9 colors at least.

question 7

(Enem 2019) A person bought a wireless device to transmit music from their computer to their bedroom radio. This device has four selector switches, each of which can be in position 0 or 1. Each choice of positions for these switches corresponds to a different transmission frequency.

The number of different frequencies that this device can transmit is determined by

a) 6.

b) 8.

c) 12.

d) 16.

e) 24

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Answer explained

For the first key there are two options, for the second key two options, as well as for the third and fourth.

Using the fundamental counting principle, there are:

2. 2. 2. 2 = 16

There are 16 different frequencies.

question 8

CONTRAN Resolutions No. 590, of 05/24/2016, No. 279, of 03/06/2018, and No. 741, of 09/17/2018, established a new standard for identification plates of Brazilian vehicles, following the rules of MERCOSUR. According to these resolutions, “Vehicle Identification Plates [...] must [...] contain 7 (seven) alphanumeric characters”. Thus, in Brazil, “the MERCOSUR license plate will have the following provision: LLLNLNN, where L is a letter and N is a number”, replacing the pre-Mercosur standard, LLLNNNN.

Assuming that there is no restriction on the characters in any of the patterns presented, how many more plaques, in relation to the old system, can be formed with the new standard of placement?

a) 16.

B)

w)

d) 24.

It is)

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Answer explained

There are 26 letter options and 10 number options. As there are no restrictions, it is possible to repeat them.

Mercosur Model LLLNLNN

Using the multiplicative principle, we have:

Pre-Mercosur Model LLLNNNN

question 9

Eduardo wants to create an email using an anagram exclusively with the seven letters that make up his name, before the @ symbol.

The e-mail will have the form *******@site.com.br and will be in such a way that the three letters “edu” always appear together and exactly in that order.

He knows that the e-mail has already been created by another user and that any other grouping of letters in his name forms an e-mail that has not yet been registered.

In how many ways can Eduardo create a desired email address?

a) 59

b) 60

c) 118

d) 119

e) 120

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Answer explained

The word E-d-u-a-r-d-o has seven letters. As the letters edu must always remain together, we have:

Edward

Constructing anagrams means shuffling the letters. In this case, we consider edu as a single block or, a letter.

edu-a-r-d-o has five elements.

For the first choice there are 5 options;

For the second choice there are 4 options;

For the third choice there are 3 options;

For the fourth choice there are 2 options;

For the fifth choice there are 1 options;

Since we want to determine the total number of options, we use the multiplicative principle.

5. 4. 3. 2. 1 = 120

However, it is necessary to remember that one of these 120 combinations is already being used by another user, who is the name eduardo.

So 120 - 1 = 119

question 10

(UFPE) A Mathematics test consists of 16 multiple-choice questions, each question having 5 alternatives, of which only one must be marked as an answer. Answering all questions at random, the number of different ways you can fill in the answer card is:

a) 80.

B) .

w) .

d)

It is)

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Answer explained

There are 5 alternatives in the 1st question It is 5 alternatives in the 2nd question It is 5 alternatives in the third question…

Thus, we have a sequence of multiplications by five with 16 factors.

5x5x5x5x... x 5

Using the power multiplication property of equal bases, we repeat the base and add the exponent. Since the exponent is 1 on each factor, the answer is:

Learn more about counting and combinatorics from:

• fundamental principle of counting
• Combinatorial analysis exercises
• Combinatorial Analysis
• Combinatorial Analysis and Probability
• Solved probability exercises (easy)

How to quote?

ASTH, Rafael. Exercises on the fundamental principle of counting.All Matter, [n.d.]. Available in: https://www.todamateria.com.br/exercicios-sobre-principio-fundamental-da-contagem/. Access at:

See too

• fundamental principle of counting
• Combinatorial Analysis Exercises
• Probability Exercises
• Solved probability exercises (easy)
• Combinatorial Analysis
• Permutation: simple and with repetition
• Combination in mathematics: how to calculate and examples
• Logical Reasoning Exercises