Exercises: Counting techniques

A restaurant offers $4$ starters, $3$ main courses and $2$ desserts. If a customer chooses their menu at random, what is the probability that they choose exactly the salad for starter, the salmon for main course and the cake for dessert?

In a $100$-meter sprint, $6$ athletes take part. If all have the same probability of winning and there are no ties, what is the probability that the finishing order is exactly A, B, C, D, E, F?

In a chess club with $10$ members, a president, a secretary and a treasurer are to be chosen at random. What is the probability that Ana is chosen as president, Carlos as secretary and Elena as treasurer?

From a group of $8$ students, $3$ are chosen at random to form a representative committee. What is the probability that the chosen ones are exactly the $3$ oldest students in the group?

A fair $6$-sided die is rolled $4$ consecutive times. What is the probability that the number $5$ is obtained on all rolls?

In a box there are $5$ red balls and $4$ blue balls. $2$ balls are drawn at random simultaneously. What is the probability that both balls are red?

All possible words, whether meaningful or not, are formed by rearranging the letters of the word "ROMA". If one of these words is chosen at random, what is the probability that it starts with the letter "M"?

$5$ fair coins are tossed simultaneously. What is the probability of getting exactly $2$ heads and $3$ tails?

Numbers with $4$ distinct digits are formed at random using the digits $1, 2, 3, 4, 5, 6$. What is the probability that the number formed is even?

In a tech company, $7$ engineers and $5$ economists are employed. A project team of $4$ people is to be chosen at random. What is the probability that the team consists of exactly $2$ engineers and $2$ economists?

From a Spanish deck of $40$ cards, $3$ cards are drawn at random without replacement. What is the probability that all three cards are of the same suit?

$5$ mathematics books and $3$ physics books are to be placed at random on a shelf. What is the probability that all the physics books are placed together?

A batch of $20$ microchips contains $4$ defective ones. A random sample of $3$ microchips is selected for inspection. What is the probability that there is at least one defective microchip in the sample?

Four friends board a bus that has $6$ stops remaining until the end of the route. If each friend is equally likely to get off at any of the stops, what is the probability that all four friends get off at different stops?

All possible words are formed by rearranging the letters of the word "BANANA". What is the probability that the three letters "A" appear together consecutively?

A multiple-choice exam consists of $10$ questions, each with $4$ answer options, only one of which is correct. If a student answers all questions at random, what is the probability that they get exactly $6$ questions right?

From a standard deck of $52$ cards (consisting of $13$ different ranks across $4$ suits), $5$ cards are drawn at random. What is the probability of getting a full house (that is, three cards of one rank and two cards of another rank)?

Eight people, including two siblings, sit at random around a circular table. What is the probability that the two siblings do not sit in adjacent seats?

$3$ distinct vertices of a regular decagon are chosen at random. What is the probability that the triangle formed by these $3$ vertices is a right-angled triangle?

$5$ distinguishable balls are distributed at random into $3$ distinguishable urns. What is the probability that none of the three urns is left empty?

A university department consists of $6$ tenured professors and $8$ associate professors. A committee of $5$ people is chosen at random. What is the probability that the committee has a majority of tenured professors (that is, $3$ or more)?

On the ground floor of a $8$-storey building, $5$ people get into the lift. If each person chooses at random the floor at which they get off (from $1$ to $8$), what is the probability that exactly two people get off on the same floor and the other three get off on floors that are different from each other and different from that of the pair?

A fair $6$-sided die is rolled $6$ consecutive times. What is the probability that all numbers from $1$ to $6$ appear exactly once, given that the first three rolls resulted exclusively in even numbers?

Four people leave their coats in a theater cloakroom. Upon leaving, due to a power cut, the attendant hands back the four coats at random. What is the probability that exactly one person receives their own coat?