Exercises: Introduction

A restaurant offers a set menu with $4$ appetizers, $3$ main courses and $5$ desserts. In how many different ways can you order a full menu consisting of an appetizer, a main course and a dessert?

Draw the tree diagram for the experiment consisting of tossing a fair coin and then rolling a tetrahedral die (with its faces numbered from $1$ to $4$). How many possible outcomes does this compound experiment have?

In a classroom library there are $12$ science fiction novels, $8$ historical novels and $5$ biographies. If you only have time to read one book this weekend, how many different options do you have to choose from?

Calculate the value of the following numerical expressions using the properties of factorials:

• $\frac{8!}{6!}$

• $4! + 3!$

A bicycle brand manufactures a model that is available in $3$ different frame sizes, $5$ colors and with $2$ different types of brakes. How many different versions of this bicycle model exist on the market?

To go from city A to city B there are $3$ different roads, and to go from city B to city C there are $2$ different roads.

• In how many ways can you travel from A to C, necessarily passing through B?

• Draw the corresponding tree diagram for this journey.

In a chest of drawers there are loose socks of $3$ different colors (black, white and blue). If there is a power outage and you take out socks in the dark, what is the minimum number of socks you must draw to ensure you have at least one pair of the same color?

Express the following products using mathematical factorial notation:

• $15 \cdot 14 \cdot 13$

• $7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$

A password for a computer system must be formed by $2$ letters of the alphabet (out of $26$ possible letters) followed by $3$ digits (from $0$ to $9$), or by $3$ letters followed by $2$ digits. If both letters and digits can be repeated, how many different passwords can be created in total?

Two basketball teams, Alpha and Beta, play a best-of-$3$ playoff series (the first to win $2$ matches qualifies). Draws are not possible. Draw the tree diagram with all possible outcomes of the playoff. In how many different ways can the series of matches unfold?

Simplify the algebraic expression $\frac{(x+2)!}{x!}$ and then solve the equation $\frac{(x+2)!}{x!} = 72$.

Using only the digits from the set $\{1, 2, 3, 4, 5\}$:

• How many numbers of $4$ different digits can be formed?

• How many of those formed numbers are even?

In a high school there are $400$ enrolled students.

• Justify mathematically why you can be sure that at least two students have their birthday on the same day of the year.

• What is the minimum number of students needed in the school to guarantee that at least three of them share a birthday? (Consider non-leap years of $365$ days).

A multiple-choice exam consists of $10$ questions. Each question has $4$ answer options, of which only one is correct.

• In how many different ways can the exam be answered if it is compulsory to mark an option for all questions?

• In how many ways could it be answered if leaving questions blank is allowed?

In an urn there are $3$ red balls and $2$ blue balls. Two balls are drawn consecutively and without returning them to the urn (without replacement). Draw the tree diagram of the experiment and calculate how many different outcomes can be obtained if we take into account the order of extraction and the color of the balls.

Look at the grid in the figure. A robot moves from point A to point B moving only to the right or upwards along the grid lines. How many different paths can the robot take to reach its destination?

We want to design a flag with $3$ vertical stripes using a palette of $5$ available colors.

• In how many ways can it be colored if two adjacent stripes cannot have the same color?

• In how many ways can it be colored if, in addition to the previous condition, the first and last stripes must also necessarily be of different colors?

Solve the following equation with factorials, knowing that $n$ is a natural number:

$5$ points are chosen at random inside a square of side $2\text{ cm}$. Prove, using the pigeonhole principle, that there will always be at least two points whose distance between them is less than or equal to $\sqrt{2}\text{ cm}$. (Hint: divide the square into strategic regions).

In the decimal number system, we consider all integers between $1000$ and $9999$ (both inclusive). How many of these numbers contain the digit $7$ at least once in their writing?

Two friends, Daniel and Diana, play a board game consisting of several rounds. The final winner of the game will be the first to win $2$ consecutive rounds or $3$ rounds in total throughout the game. Knowing that draws are not possible in any round, draw the corresponding tree diagram and determine the maximum number of rounds they can play and how many different outcomes the game has.

In a class of $20$ students, a management team consisting of a president, a vice president, and a secretary must be chosen (different roles that cannot be held by the same person simultaneously). Once the management team is chosen, a sports representative must be selected from the remaining $17$ students. In how many different ways can this entire selection process be carried out?

A palindromic number is one that reads the same from left to right as from right to left (for example, the number $35153$).

• How many palindromic numbers of $5$ digits exist in total?

• How many of those palindromic numbers of $5$ digits are even numbers?

Prove algebraically the following identity valid for any natural number $n \ge 1$:

Once proven, apply this result to simplify and calculate the exact value of the expression $\frac{10! + 9!}{11!}$.