Exercises: Combinatorics

A restaurant offers $4$ starters, $5$ main courses and $3$ desserts on its set menu. Calculate how many different complete meals (consisting of a starter, a main course and a dessert) a customer can order.

Calculate how many different ways $6$ different textbooks can be arranged in a line on a shelf.

A robotics club has $12$ members. Determine how many different ways a committee consisting of a president, a vice president and a secretary can be chosen, given that no person can hold more than one position.

A technology teacher needs to select $4$ students from a class of $20$ to attend a science fair. Calculate how many different groups of representatives can be formed.

Calculate the number of different anagrams, whether meaningful or not, that can be formed using all the letters of the word $INGENIERIA$.

A digital bicycle lock is opened using a $4$-digit code (from $0$ to $9$). Determine how many different codes can be set if repeating digits is allowed.

$8$ points are marked on a plane, no three of which are collinear. Determine how many different straight lines can be drawn by connecting pairs of these points.

From a Spanish deck of $40$ cards, $3$ cards are drawn at random without replacement. Calculate the probability that all three cards are Kings.

Seven friends go to the cinema and sit in a row of $7$ seats. Two of them, Ana and Bruno, must sit next to each other. Calculate how many different ways they can occupy the seats.

A biomedical research team must consist of $5$ people. The available candidates are $6$ biologists and $4$ chemists. Determine how many different teams can be formed if the team must contain exactly $3$ biologists and $2$ chemists.

A batch of $20$ microchips contains $4$ defective ones. If $3$ microchips are selected at random without replacement for inspection, calculate the probability that exactly $1$ of them is defective.

A security system requires a password consisting of $2$ distinct vowels followed by $3$ distinct digits (from $0$ to $9$). Calculate how many different passwords can be configured under these conditions.

The letters of the word $AEROPUERTO$ are shuffled at random to form a new anagram. Calculate the probability that the new word starts and ends with the letter $O$.

In a chess tournament where each participant plays exactly one match against every other player, a total of $105$ matches were played. Determine the number of players who participated in the tournament.

From a standard deck of $52$ cards, a hand of $5$ cards is dealt. Calculate the probability that the player gets exactly $3$ Aces in their initial hand.

An urn contains $5$ red balls, $4$ blue balls and $3$ green balls. If $3$ balls are drawn simultaneously at random, calculate the probability that they are all of different colors.

A tech company is going to form a cybersecurity committee of $6$ experts chosen from $8$ software engineers and $7$ network analysts. Calculate how many different committees can be formed if there must be at least $4$ software engineers in the group.

Solve the following combinatorial equation to find the natural value of $n$:

$3 \times V_{n, 3} = 4 \times V_{n-1, 3}$

Four men and four women are sitting down to dinner around a circular table. Determine how many different ways they can be seated if people of the same gender cannot sit next to each other (that is, men and women must alternate). Consider two arrangements to be identical if one can be obtained by rotating the other.

In a quality control process, a random sample of $5$ items is taken from a production line containing a total of $50$ items, $10\%$ of which are defective. Calculate the probability of finding at most $1$ defective item in the sample taken.

An investor wishes to divide $200\,000$ pesos among $4$ startups. If the investment in each company must be an exact multiple of $20\,000$ pesos (including the possibility of investing $0$ pesos in any of them), determine how many different ways the capital can be distributed.

$5$ cards are drawn at random from a standard deck of $52$ cards. Calculate the probability of obtaining a full house, that is, a hand consisting of $3$ cards of one rank and $2$ cards of another rank.

In a city whose streets form a perfect grid, calculate how many paths of minimum length exist to go from point $A(0,0)$ to point $B(6,5)$, passing through intersection $C(3,2)$.

The European Space Agency needs to assign $12$ astronauts to $3$ different space modules (Command Module, Lunar Module and Service Module). Calculate how many different ways the crew members can be distributed if each module must accommodate exactly $4$ astronauts.