Introduction to Combinatorics

A restaurant offers a set menu with $4$ starters, $3$ main courses and $5$ desserts. How many different ways can a full three-course meal consisting of a starter, a main course and a dessert be ordered?

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 are on the market?

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?

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, ensuring you pass through B?

• Draw the tree diagram corresponding to this journey.

You can do it on paper and send a photo with your answer.

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

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

• In how many different ways can the exam be answered if selecting an option for every question is mandatory?

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

A flag with $3$ vertical stripes is to be designed 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 be of different colors?

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?

You can do it on paper and send a photo with your answer.

Two basketball teams, Alfa and Beta, play a best-of-$3$ playoff (the first team to win $2$ games 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 games unfold?

You can do it on paper and send a photo with your answer.

An urn contains $3$ red balls and $2$ blue balls. Two balls are drawn sequentially without replacement. Draw the tree diagram for the experiment and calculate how many distinct outcomes can be obtained if the order of extraction and the color of the balls are taken into account.

Two friends, Ionut and Diana, play a board game consisting of several rounds. The overall winner of the game will be the first person to win $2$ consecutive rounds or $3$ rounds in total across the match. 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 game paths there are.

You can do it on paper and send a photo with your answer.

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

• How many numbers with $4$ distinct digits can be formed?

• How many of these formed numbers are even?

In the decimal number system, consider all integers from $1000$ to $9999$ (inclusive). How many of these numbers contain the digit $7$ at least once in their representation?

A palindrome number is a number that reads the same forwards and backwards (for example, the number $35153$).

• How many $5$-digit palindrome numbers are there in total?

• How many of these $5$-digit palindrome numbers are even numbers?

In a class of $20$ students, a leadership team consisting of a president, a vice-president, and a secretary must be chosen (distinct positions that cannot be held by the same person simultaneously). Once the leadership team has been 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?

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?

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

• $\frac{8!}{6!}$

• $4! + 3!$

Express the following products using factorial notation:

• $15 \times 14 \times 13$

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

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

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

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 $\frac{10! + 9!}{11!}$.

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 pull out to guarantee having at least one pair of the same color?

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

• Justify mathematically why you can guarantee that at least two students share the same birthday.

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

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