Task 6 (the problem of exiting a vehicle). In an elevator on the first floor of a twelve-story building, 6 passengers entered. In how many ways can they exit the elevator on the required floors, starting from the third?
Solution. For one of the passengers, there are 10 possibilities to exit the elevator on one of the 10 floors (from the third to the twelfth inclusive). The other passenger also has 10 possibilities to exit the elevator. The same can be said about the other passengers. Thus, the multiplication rule applies, according to which the number of ways the passengers can exit the elevator is $10^{6}$.

Answer. 1,000,000 ways.

## Exercises

1. How many different lists can be formed from five people, with five in each list?
2. How many different words (including meaningless ones) can be formed from seven different letters?
3. In how many ways can 5 men and 5 women be seated at a table so that people of the same gender do not sit next to each other?
4. How many different five-digit numbers can be formed using the digits $1,2,3,5,7$ (no digit should be repeated)?
5. How many different five-digit numbers can be formed using the digits $0,1,2,3,5,7$ (no digit should be repeated)?
6. In a room, there are 6 chairs. In how many ways can: a) 6 guests be seated? b) 4 guests be seated?
7. In a football championship, 21 teams are participating. In how many ways can the gold, silver, and bronze medals be distributed?
8. The school administration decided to reward a group of 25 students with five different-cost vacation trips. In how many ways can this be implemented?
9. A team of 5 people is participating in a running competition with 15 other athletes. In how many ways can the places occupied by the members of this team be distributed?
10. A committee consists of a chairman, his deputy, a secretary, and 12 other members. In how many ways can the members of the committee distribute their responsibilities among themselves?
11. In a group of 25 people, 20 need to be selected for a certain task. In how many ways can this selection be made?
12. There is a group of 10 soldiers and 7 officers. In how many ways can a patrol group consisting of 5 soldiers and 2 officers be formed?
13. From a vase containing 11 red and 6 pink carnations, 3 red and 2 pink flowers are selected. In how many ways can this be done?
14. Andrei has 5 different history books, and Oleg has 7 geography books. In how many ways can they exchange three books with each other?
15. In how many ways can 4 balls be drawn (simultaneously) from a box containing 21 balls?
16. In how many ways can a ten-volume collection of Jack London's works be arranged on a bookshelf:
1) in any order;
2) so that the first three volumes are together (in any order);
3) so that the first three volumes are not together (in any order)?
17. In a vase, there are 10 red and 4 pink carnations. In how many ways can three flowers of different colors be selected from the vase?
18. How many different six-digit numbers can be formed from the digits $1,2,3,4,5,6,7$ such that the digits do not repeat and the extreme digits are even?
19. In a box, there are 20 parts, 7 of which are defective. A set of 6 parts is randomly selected. How many sets are there in which 3 parts are defective? How many sets are there in which no more than 3 parts are defective?
20. From a group of 20 people, 4 need to be selected for one task and 3 for another. In how many ways can this be done?

## Answers

(answers given in symbolic form contain implicit hints to their solution)

1. 5!. 2. 7!. 3. $2(5 \text { !) })^{2}$. 4. 5!. 5. 5 . 5!. 6. a) 6!; b) 360. 7. 7980. 8. $A_{25}^{5}$. 9. $A_{20}^{5}$. 10. $A_{15}^{3}, 11 . C_{25}^{5}, 12 . C_{10}^{5} C_{7}^{2}$. 13. $C_{11}^{3} C_{6}^{2}$, 14. $C_{5}^{3} C_{7}^{3}$, 15. $C_{21}^{4}$. 16. 1) $10!=3628800$; 2) $8!3!=241920$; 3) $10!-8!3!=3386880$. 17. $C_{10}^{2} C_{4}^{1}+C_{10}^{1} C_{4}^{2}=$ 240. 18. $A_{3}^{2} \cdot A_{5}^{4}=720$. 19. $C_{7}^{3} C_{13}^{3} ; C_{13}^{6}+C_{7}^{1} C_{13}^{5}+$ $+C_{7}^{2} C_{13}^{4}+C_{7}^{3} C_{13}^{3} . \mathbf{2 0 .} C_{20}^{3} C_{17}^{4}$.

## § 2. Basic Concepts of Probability Theory

$1^{\circ}$. An experiment, or trial, refers to the practical implementation of a set of conditions under which the phenomenon under study can be observed.

The result of the trial (experiment) is called an event, or outcome of the trial.

Events are denoted by capital letters $A, B, C, \ldots$.

A certain event is one that will definitely occur as a result of the trial. An impossible event is one that will definitely not occur in the given trial.

An event $A$ is called random if, as a result of the trial, it may or may not occur.