Omnibus N 9-10 2007.
Marine soul of route lights.
Tradition is a mysterious thing. At first it is carefully observed, trying to maintain all the nuances, it is brought to the point of superstition, then suddenly they discover that it does not live up to the expectations placed on it, does not meet logic, has no scientific justification- and they break with tradition, and subsequently notice with sadness that with its loss something beautiful and necessary has gone away. . .
Even quite recently, there was a tradition of giving tram routes not only a digital, but also a color designation - route lights were lit on both sides of the route number, in front and behind the car. Streets with tram traffic were distinguished by a special, festive elegance; drivers, passengers, track workers, dispatchers and switchmen navigated the tram flow using route lights; many could not imagine a tram without colored lights. The Moscow system of route lights was built on a unique correspondence between numbers and colors. “1” is always red, “2” is green, “5” is olive, “7” is blue, and so on. But in Leningrad the lights “spoke” in another language,
and reading them “in Moscow” most often led to nonsense, since there were not 10 lights, as in Moscow, but only five. They were well differentiated, and their combinations always looked very beautiful. However, out of five lights, 25 different combinations of two are possible, while the routes in St. Petersburg-Leningrad eventually became about 70, so the route signs could be repeated. For example, two whites - 9, 43; red and yellow - 1, 51, 64; blue and red - 33, 52, 54; two red ones - 5, 36, 39, 45, 47. And only route N 20 was designated the same by the Moscow and St. Petersburg systems: green and white.
It happened that the route lights in St. Petersburg changed. If it happened that after changing one of the routes, it worked on a fairly long section with another route having the same colors, then the composition of the lights for one of these routes had to be changed.
Route N 4 used to run from Dekabristov Island to the Volkov Cemetery and was marked by two yellow (orange) lights. Then the route was closed and opened under the same number in another place with other lights: blue + blue, since it shared a section with the 35th tram (two yellow).
Route N 43 initially had lights: red + white. When extended to the port in 1985, the lights changed: white + white, as the route began to share a section with tram N 28 (red + white). Route 3 was marked with green and white colors. When the lights were restored in 2007, the combination was replaced with yellow + green. At the same time, the combinations changed on a number of other routes: 48 (was: white + white, now: blue + blue); 61 (was: white + white, now: white + yellow), etc.
The St. Petersburg system of route lights, so simple in appearance and so intricate, is associated with the tradition of primarily European tram cities. Thus, already in 1907, a letter to the newspaper “Novoe Vremya” contained a request from “ordinary people Vasilyevsky Island"introduce colored lights on trams, "like abroad, in particular in Frankfurt am Main." At present, remnants of the former systems have been preserved in the form of colored diagonal lighting on tram route signs in Amsterdam. This tradition, in turn, is probably , rises to the lights marine navigation. Why specifically to the sea, and not, say, to the railway? Yes, because route lights, like sea lights, do not prohibit or force anyone to do anything, but simply help them find their way in the dark.
Marine navigation lights are deciphered in special maritime books - sea directions. Route lights are also described in city guides. The first of them was the “Mobile Guide to St. Petersburg Trams,” published by the publishing house E.I. Marcus (1910).
The composition of the colors used in St. Petersburg route lights (white, red, orange or yellow, green, blue) differs little from the colors of sea lights (white, red, orange, green, blue, purple).
If you look closely, you can find other similarities, but it is much more important to understand why such a lax system of route lights, requiring constant adjustment, has taken root in prudent St. Petersburg. The answer is simple: after all, St. Petersburg is a seaside city, and it equally Characterized by both the severity of architectural forms and the frivolity of the carnival, and therefore the cheerful colors of the route lights.
In 2007, the tradition came to new round. LED route lights are now installed on carriages. They will shine not only in the evening twilight, but also in daylight.
Orenburg 250 300 200 300 600 Order 600 500 200 100 c1 = 250; c2 = 200; с3 = 150. b) Table 22 Branches Moscow St. Petersburg Tver Tula Purchase volume Supplier Gdansk 200 300 250 150 550 Krasnodar 300 400 300 250 650 Orenburg 150 250 200 200 800 Order 450 700 3 00 300 c1 = 200; c2 = 100; c3 = 150. c) Table 23 Branches Moscow St. Petersburg Tver Tula Purchase volume Supplier Gdansk 200 300 250 150 650 Krasnodar 250 400 300 250 750 Orenburg 150 250 200 200 600 Order 500 750 4 00 300 c1 = 200; c2 = 100; c3 = 150. Problem 2. Four stores “Liga-plus”, “Umka”, “Gurman” and “Uley” sell dairy products supplied by three dairies. The first plant has an agreement with the Gurman brand store on a fixed supply of its products. Tariffs for the delivery of dairy products and the volume of fixed delivery (in boxes) are given in the tables by option. Find the optimal plan for the supply of dairy products. a) Table 24 Store "Liga-plus" "Gourmand" "Umka" "Beehive" Purchase volume Plant 1 5 8 6 10 700 200 2 9 6 7 5 800 3 6 7 5 8 500 800 400 600 200 b) Table 25 Store “Liga-plus” “Gourmet” “Umka” “Beehive” Purchase volume Plant 1 5 10 7 400 300 5 2 6 8 5 8 600 3 7 9 6 4 900 500 700 200 500 TO THE SECTION “COMBINATORICS” Task 3 Table 26 Option Task No. I a) The commission consists of a chairman, his deputy and five more people. In how many ways can members of the commission distribute responsibilities among themselves? b) The championship, in which 16 teams participate, is held in two rounds (i.e., each team meets every other team twice). Determine how many meetings should be held. c) Two rooks of different colors are placed on the chessboard so that each can take the other. How many such locations are there? II a) In how many ways can you select three officers on duty from a group of 20 people? b) The lock opens only if a certain three-digit number is dialed. The attempt consists of dialing three digits at random out of the given five digits. It was possible to guess the number only on the last of all possible attempts. How many attempts preceded the successful one? c) The order of performance of the eight participants in the competition is determined by lot. How many different outcomes of the draw are possible? III a) How many different sound combinations can be used on ten selected piano keys, if each sound combination can contain from three to ten sounds? b) From a group of 15 people, four participants in the 800 + 400 + 200 + 100 relay are selected. In how many ways can athletes be arranged according to the stages of the relay? c) A bookshelf holds 30 volumes. In how many ways can they be arranged without the first and second volumes standing next to each other? IV a) There are 10 red and 5 pink carnations in a vase. In how many ways can you select five carnations of the same color from a vase? b) A team of five people competes in a swimming competition in which 20 other athletes participate. In how many ways can the places occupied by the members of this team be distributed? c) The metro train makes 16 stops at which all passengers get off. In how many ways can 100 passengers who boarded the train at the final stop be distributed between these stops? Continuation of the table. 26 Option V a) Numbers tram routes sometimes indicated by two colored lights. How many different routes can be marked if eight colors of lights are used? b) In how many ways can two rooks be placed on a chessboard so that one cannot capture the other? (One rook can take another if it is on the same horizontal or vertical of the chessboard). c) How much three-digit numbers divisible by 3 can be made up of the numbers 0, 1, 2, 3, 4, 5, if each number must not contain identical numbers? TO THE SECTION “PROBABILITY THEORY”: Task 4 Table 27 Task Option a) Classic and statistical definition probabilities I Two thrown dice. Find the probability that the sum of points on the rolled sides is even, and a six appears on the side of one of the dice. II When transporting a box that contained 21 standard and 10 non-standard parts, one part was lost, and it is not known which one. The part removed at random (after transporting the box) turned out to be standard. Find the probability that the following was lost: a) a standard part; b) non-standard part III A cube, all edges of which are painted, is sawn into a thousand cubes of the same size, which are then thoroughly mixed. Find the probability that a cube drawn at random has: a) one colored face; b) two painted edges; c) three colored faces IV In the envelope, among 100 photographs, there is one wanted one. 10 cards are drawn at random from the envelope. Find the probability that the desired one will be among them V There are five identical parts in the box, three of them are painted. Two items were removed at random. Find the probability that among two extracted products there will be: a) one painted product; b) two painted products; c) at least one painted product b) Theorems of addition and multiplication of probabilities I 15 textbooks are randomly arranged on a library shelf, 5 of them are bound. The librarian selects three textbooks at random. Find the probability that at least one of the taken textbooks will be bound. Continuation of the table. 27 Option of Task II There are 10 parts in a box, 4 of which are painted. The assembler took 3 parts at random. Find the probability that at least one of the parts taken is painted III To signal an accident, two independently operating alarms are installed. The probability that the first alarm will go off during an accident is 0.95, and the probability that the second alarm will go off during an accident is 0.9. Find the probability that during an accident only one alarm will go off. IV Two shooters are shooting at a target. The probability of hitting the target with the first shot for the first shooter is 0.7, and for the second - 0.8. Find the probability that during the first salvo only one of the shooters will hit the target V From the batch, the merchandiser selects the highest grade products. The likelihood that that a product taken at random will be of the highest grade is 0.8. Find the probability that out of three inspected products only two products are of the highest grade c) Probability of the occurrence of at least one event I B electrical circuit three elements are connected in series, operating independently of one another. The failure probabilities of the first, second and third elements, respectively, are equal to p1 = 0.1; p2 = 0.15; p3 = 0.2, find the probability that there will be no current in the circuit II The device contains two independently operating elements. The failure probabilities of elements are 0.05 and 0.08, respectively. Find the probability of device failure if it is enough for at least one element to fail. III To destroy the bridge, it is enough to be hit by one aerial bomb. Find the probability that the bridge will be destroyed if four bombs are dropped on it, the probabilities of which are respectively equal to: 0.3; 0.4; 0.6; 07 IV The probability of at least one shooter hitting the target with three shots is 0.875. Find the probability of a hit with one shot V Probability successful implementation exercises for each of the two athletes is 0.5. Athletes perform the exercise in turn, each making two attempts. The first person to complete the exercise receives a prize. Find the probability of athletes receiving a prize d) Formula full probability I Dropped into an urn containing two balls white ball, after which one ball is randomly drawn from it. Find the probability that the extracted ball will be white if all possible assumptions about the initial composition of the balls (based on color) are equally possible. Table continued. 27 End of Tab Option of Task II There are five rifles in the pyramid, three of which are equipped optical sight . The probability that a shooter will hit the target when firing from a rifle with an optical sight is 0.95; for a rifle without an optical sight, this probability is 0.7. Find the probability of being hit if the shooter fires one shot from a rifle taken at random. III The first urn contains 10 balls, of which 8 are white, the second urn contains 20 balls, of which 4 are white. One ball was drawn at random from each urn, and then one ball was drawn at random from these two balls. Find the probability that a white ball is drawn IV Each of the three urns contains 6 black balls and 4 white balls. One ball is randomly drawn from the first urn and placed into the second urn, after which one ball is randomly drawn from the second urn and placed into the third urn. Find the probability that a ball drawn at random from the third urn turns out to be white V The box contains 12 parts manufactured at plant 1, 20 parts manufactured at plant 2 and 18 parts manufactured at plant 3. The probability that the part manufactured at plant 1 of excellent quality, equal to 0.9; for parts manufactured at factories 2 and 3, these probabilities are 0.6 and 0.9, respectively. Find the probability that a part extracted at random will be of excellent quality e) Basic formulas of probability theory I There are 10 rifles in the pyramid, 4 of which are equipped with an optical sight. The probability that a shooter will hit a target when firing a rifle with a telescopic sight is 0.95; for a rifle without an optical sight, this probability is 0.8. The shooter hit the target with a rifle taken at random. What is more likely: the shooter shot from a rifle with or without an optical sight? II On average, 50% of patients with disease A are admitted to a specialized hospital, 30% with disease B, 20% with disease C. The probability of a complete cure for disease A is 0.7; for diseases B and C these probabilities are 0.8 and 0.9, respectively. The patient admitted to the hospital was discharged healthy. Find the probability that this patient suffered from disease A III. Two equal opponents are playing chess. What is more likely: a) to win one game out of two or two games out of four; b) win at least two games out of four or at least three games out of five? Nobody's records are taken into account IV There are five children in the family. Find the probability that among these children: a) two boys; b) no more than two boys; c) more than two boys; d) no less than two and no more than three boys. The probability of having a boy is taken to be 0.51 V. A coin is tossed five times. Find the probability that heads will appear: a) less than twice; b) at least twice Task 5 Table 28 Option Task a) Discrete random variables, numerical characteristics of discrete random variables I 1.1 Discrete random variable X is given by the distribution law X 0.1 0.3 0.6 0.8 P 0.2 0 ,1 0.4 0.3 Construct a distribution polygon. 1.2 The textbook was published in a circulation of 100,000 copies. The probability that the textbook is bound incorrectly is 0.0001. Find the probability that the circulation contains five defective books. 1.3 For a discrete random variable X from section 1.1. find: a) mathematical expectation and variance; b) initial moments of the first, second and third orders; c) central moments of the first, second, third and fourth orders. 1.4 Using Chebyshev’s inequality, estimate for the discrete random variable X from section 1.1 the probability that │ X – M(X) │< 0,2
II 1.1 Дискретная случайная величина X задана законом распределения
X 0,10 0,15 0,20 0,25
P 0,1 0,3 0,2 0,4
Построить многоугольник распределения.
1.2 Устройство состоит из 1000 элементов, работающих независимо
один от другого. Вероятность отказа любого элемента в момент вре-
мени Т равна 0,002. Найти вероятность того, что за время Т откажут
ровно три элемента.
1.3 Для дискретной случайной величины X из п. 1.1 найти: а) мате-
матическое ожидание и дисперсию; б) initial moments first, second and third orders; c) central moments of the first, second, third and fourth orders. 1.4. Using Chebyshev’s inequality, estimate for the discrete random variable X from Section 1.1 the probability that │ X – M(X) │< 0,7
III 1.1 Дискретная случайная величина X задана законом распределения
X 0,2 0,4 0,5 0,6
P 0,3 0,1 0,2 0,4
Построить многоугольник распределения.
1.2 Станок штампует детали. Вероятность того, что изготовленная
деталь окажется бракованной, равна 0,01. Найти вероятность того, что
среди отобранных 200 деталей окажется ровно 4 бракованных.
1.3 Для дискретной случайной величины X из п. 1.1. найти: а) мате-
матическое ожидание и дисперсию; б) начальные моменты первого,
второго и третьего порядков; в) центральные моменты первого, второ-
го, третьего и четвертого порядков.
1.4 Используя неравенство Чебышева, оценить для дискретной слу-
чайной величины X из п. 1.1 вероятность того, что │ X – M(X) │< 0,5
Продолжение табл. 28
Вариант Задание
IV 1.1 Дискретная случайная величина X задана законом распределения
X 0,2 0,6 0,9 1,2
P 0,3 0,1 0,2 0,4
Построить многоугольник распределения.
1.2 Завод направил на базу 500 изделий. Вероятность повреждения
изделия в пути равна 0,002. Найти вероятности того, что в пути будет
повреждено изделий: а) ровно 3; б) менее трех; в) более трех; г) хотя
бы одно.
1.3 Для дискретной случайной величины X из п. 1.1 найти: а) мате-
матическое ожидание и дисперсию; б) начальные моменты первого,
второго и третьего порядков; в) центральные моменты первого, второ-
го, третьего и четвертого порядков.
1.4. Используя неравенство Чебышева, оценить для дискретной слу-
чайной величины X из п. 1.1 вероятность того, что │ X – M(X) │< 0,6
V 1.1 Дискретная случайная величина X задана законом распределения
X 0,3 0,4 0,7 0,10
P 0,4 0,1 0,2 0,3
Построить многоугольник распределения.
1.2 Магазин получил 1000 бутылок mineral water. The probability that the bottle will be broken is 0.003. Find the probability that the store will receive broken bottles: a) exactly 2; b) less than two; c) more than two; d) at least one. 1.3 For a discrete random variable X from clause 1.1, find: a) mathematical expectation and variance; b) initial moments of the first, second and third orders; c) central moments of the first, second, third and fourth orders. 1.4 Using Chebyshev’s inequality, estimate for the discrete random variable X from section 1.1. probability that │ X – M(X) │< 0,1
б) Непрерывные случайные величины, числовые характеристики непрерыв-
ных случайных величин, распределения непрерывной случайной величины.
I 1.1 Дана функция распределения непрерывной случайной величины X
0, x ≤ 0;
F(X)= sin x, 0 < x ≤ Π /2;
1, x >Π/2. Find the distribution density f(x). 1.2 Random value X is specified by the distribution density f(x) = 2x on the interval (0; 1); outside this interval f(x) = 0. Find the mathematical expectation and variance of the value X. 1.3 The random variable X is specified by the distribution density f(x) = 0.5x in the interval (0; 2), outside this interval f(x) = 0. Find the initial and central moments of the first, second, third and fourth orders. 1.4 Find the dispersion and standard deviation of a random variable X, distributed uniformly in the interval (2; 8) Continuation of table. 28 Option Task II 1.1 Given the distribution function of a continuous random variable X 0, x ≤ 0; F(X) = sin 2x, 0< x ≤ Π /4;
1, x >Π/4. Find the distribution density f(x). 1.2 The random variable X is specified by the distribution density f(x) = (1/2)x on the interval (0; 2); outside this interval f(x) = 0. Find the mathematical expectation and variance of the value X. 1.3 The random variable X is given by the distribution density f(x) = 2x in the interval (0; 1), outside this interval f(x) = 0 Find the initial and central moments of the first, second, third and fourth orders. 1.4 Random variables X and Y are independent and distributed uniformly: X in the interval (a, b), Y in the interval (c, d). Find the mathematical expectation and variance of the product XY III 1.1 Given the distribution function of a continuous random variable X 0, x≤0; F(X) = cos 2x, 0
Previously, tram numbers were indicated by two colored lanterns. How many different routes can be marked using eight lights? various colors?
Answers:
the formula will be: 8²=64 64 different routes.
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Combinatorics problems
| Parameter name | Meaning |
| Article topic: | Combinatorics problems |
| Rubric (thematic category) | Mathematics |
1. One day's schedule contains 5 lessons. Determine the number of such schedules when choosing from eleven disciplines.
Answer: 55,440.
2. The commission consists of a chairman, a deputy and five more people. In how many ways can committee members distribute responsibilities among themselves?
Answer: 42.
3. In how many ways can you select three officers on duty from a group of 20 people?
Answer: 1 140.
4. How many different sound combinations can be played on ten selected piano keys, if each sound combination can contain from three to ten sounds?
Answer: 968.
5. There are 10 red and 5 pink carnations in a vase. In how many ways can you select five carnations of the same color from a vase?
Answer: 253.
6. Tram route numbers are sometimes indicated by two colored lights. How many different routes can be marked if eight colors of lanterns are used?
Answer: 64.
7. The championship, which features 16 teams, is played in two rounds (i.e. each team plays every other team twice). Determine how many meetings should be held.
Answer: 240.
8.
The lock opens only if a certain three-digit number is dialed.
Posted on ref.rf
The attempt consists of dialing three digits at random out of the given five digits.
Posted on ref.rf
It was possible to guess the number only on the last of all possible attempts. How many attempts preceded the successful one?
Answer: 124.
9. From a group of 15 people, four participants in the 800+400+200+100 relay are selected. In how many ways can athletes be arranged according to the stages of the relay?
Answer: 32,760.
10. A team of five competes in a swimming competition with 20 other athletes competing. In how many ways can the places occupied by the members of this team be distributed?
Answer: 25!/20!.
11. In how many ways can two rooks be placed on a chessboard so that one cannot capture the other? (One rook can take another if it is on the same horizontal or vertical line of the chessboard.)
Answer: 3 126.
12. Two rooks of different colors are placed on the chessboard so that each can capture the other. How many such locations are there?
Answer: 896.
13. The order of performance of the eight participants in the competition is determined by lot. How many different outcomes of the draw are possible?
The answer is ˸ 8!.
14. Thirty people are divided into three groups of ten people each. How much should it be various compositions groups?
Answer˸ 30!/(10!).
15. How many four-digit numbers divisible by 5 can be made from the digits 0, 1, 3, 5, 7, if each number must not contain the same digits?
Answer: 42.
16. How many different glowing rings can be made by placing 10 different-colored light bulbs around a circle (the rings are considered the same if the colors are in the same order)?
The answer is ˸ 9!.
17. The bookshelf holds 30 volumes. In how many ways can they be arranged without the first and second volumes standing next to each other?
18. Four shooters must hit eight targets (two each). In how many ways can they distribute the targets among themselves?
Combinatorics problems - concept and types. Classification and features of the category "Problems in Combinatorics" 2015, 2017-2018.
set of vectors (b n ) there is a bijection (prove it!). Hence,
C n m (n) is equal to the number of vectors b n. “The length of the vector”b n is equal to the number 0 and 1, or m + +n–
1. The number of vectors is equal to the number of ways in which m units can be placed in m +n 1 places, and this will be C n m +m- 1 .
Example 9. There are 7 types of cakes in a pastry shop. The buyer takes 4
cakes. In how many ways can he do this? (It is assumed that
cakes of each type 4). | ||||||
The number of ways will be C 4 | 210. |
|||||
7+ 4- 1 | 4! 6! 1 2 3 4 |
|||||
Example10. Let V = (a,b,c). Sample size m = 2. List permutations, placements, combinations, placements with repetitions, combinations with repetitions.
1. Permutations: ( abc ,bac ,bca ,acb ,cab ,cba ).P 3 =3!=6.
2. Placements: ((ab), (bc), (ac), (ba), (cb), (ca)).A 3 2 1 3 ! ! 6.
3. Combinations: ((ab), (ac), (bc)).C 2 | ||||||||||
1! 2! | ||||||||||
4. Placements with repetitions: ((ab), (bc), (ac), (ba), (cb), (ca), (aa), (bb), |
||||||||||
(cc)). | (3)= 32 | |||||||||
Combinations | with repetitions: | ((ab), | (bc), (ca), (aa), (bb), (cc)). |
|||||||
C2(3)C2 | ||||||||||
3+ 2- 1 | ||||||||||
1.2. Combinatorics problems
1. One day's schedule contains 5 lessons. Determine the number of such schedules when choosing from eleven disciplines.
Answer: 55,440.
2. The commission consists of a chairman, his deputy and five more people.
In how many ways can committee members distribute responsibilities among themselves?
3. In how many ways can you select three duty officers from a group of 20
Answer: 1,140.
4. How many different sound combinations can be played on ten selected piano keys, if each sound combination can contain from three to ten sounds?
Answer: 968.
5. There are 10 red and 5 pink carnations in a vase. In how many ways can you select five carnations of the same color from a vase?
Answer: 253.
6. Tram route numbers are sometimes indicated by two colored lights. How many different routes can be marked if eight colors of lanterns are used?
7. The championship, in which 16 teams participate, is held in two rounds (i.e.
each team plays every other team twice). Determine how many meetings should be held.
Answer: 240.
8. The lock opens only if a certain three-digit number is dialed. The attempt consists of dialing three digits at random out of the given five digits. It was possible to guess the number only on the last of all possible attempts. How many attempts preceded the successful one?
Answer: 124.
9. From a group of 15 people, four relay participants are selected
800+400+200+100. In how many ways can athletes be arranged according to the stages of the relay?
Answer: 32,760.
10. A team of five people competes in a swimming competition,
in which 20 more athletes participate. In how many ways can the places occupied by the members of this team be distributed?
Answer: 25!/20!.
11. In how many ways can two rooks be placed on a chessboard so that one cannot capture the other? (One rook can take another,
if she is on the same horizontal or vertical chessboard.)
Answer: 3,126.
12. Two rooks of different colors are placed on a chessboard so that each can take the other. How many such locations are there?
Answer: 896.
13. The order of performance of the eight participants in the competition is determined by lot. How many different outcomes of the draw are possible?
14. Thirty people are divided into three groups of ten people each.
How many different group compositions can there be?
Answer: 30!/(10!) 3.
15. How many four-digit numbers divisible by 5 can be made from the digits 0, 1, 3, 5, 7, if each number must not contain the same digits?
16. How many different glowing rings can be made by placing 10 different-colored light bulbs around a circle (the rings are considered the same if the colors are in the same order)?
17. A bookshelf holds 30 volumes. In how many ways can they be arranged without the first and second volumes standing next to each other?
Answer: 30! 2 29!.
18. Four shooters must hit eight targets (two each). In how many ways can they distribute the targets among themselves?
Answer: 2,520.
19. From a group of 12 people, two people on duty are selected every day for 6 days. Determine quantity various lists on duty if each person is on duty once.
Answer: 12!/(2!) 6.
20. How many four-digit numbers made up of the digits 0, 1, 2, 3, 4, 5 contain the digit 3 (digits are not repeated in numbers)?
Answer: 204.
21. Ten groups study in ten consecutive classrooms. How many scheduling options are there in which groups No. 1 and No. 2 would be in adjacent classrooms?
Answer: 2 9!.
22. 16 chess players are participating in the tournament. Determine the number of different schedules of the first round (schedules are considered different if the participants in at least one game differ; the color of the pieces and the board number are not taken into account).
Answer: 2,027,025.
23. Six boxes various materials delivered to five floors of the construction site. In how many ways can materials be distributed among floors? In how many variants is it delivered to the fifth floor? any one material?
Answer: 56 ; 6 45.
24. Two postmen must deliver 10 letters to 10 addresses. How many
ways they can distribute the work? Answer: 210.
25. The metro train makes 16 stops, at which all passengers get off. In how many ways can 100 passengers boarding the train at the final stop be distributed between these stops?
Answer: 16100.
26. How many three-digit numbers divisible by 3 can be made from the digits 0, 1, 2, 3, 4, 5, if each number must not contain the same digits?
27. The meeting of 80 people elects a chairman, a secretary and three members of the audit commission. In how many ways can this be done?
Answer: 80!(3! 75!).
28. Of the 10 female tennis players and 6 tennis players, 4 mixed doubles are made up. In how many ways can this be done?
Answer: 10!/48.
29. Three vehicles No. 1, 2, 3 must deliver goods to six stores. In how many ways can machines be used if the carrying capacity of each of them allows them to take goods for all stores at once and if two machines
V the same store are not sent? How many route options are possible if you decide to use only car No. 1?
Answer: 3 6 6!.
30. Four boys and two girls choose the sports section. Only boys are accepted into the hockey and boxing section; rhythmic gymnastics- only girls, and in the skiing and skating sections - both boys and girls. In how many ways can these six people be distributed among the sections?
Answer: 2304.
31. From a laboratory that employs 20 people, 5 employees must go on a business trip. How many different compositions of this group can there be?
if the head of the laboratory, his deputy and Chief Engineer Shouldn't they leave at the same time?
Answer: 15,368.
32. There are 10 people studying in a piano club; artistic word–15, in the vocal circle – 12, in the photography circle – 20 people.
In how many ways can a team of four readers, three pianists, five singers and one photographer be formed?
Answer: 15!10/7!
33. Twenty-eight dominoes are distributed among four players. How many different distributions are possible?
Answer: 28!/(74 .!}
34. From a group of 15 people, a foreman and 4 team members should be selected. In how many ways can this be done?
Answer: 15,015.
35. Five students should be divided into three parallel classes.
In how many ways can this be done? Answer: 35.
36. The elevator stops at 10 floors. In how many ways can 8 passengers in the elevator be distributed between these stops?
Answer: 108.
In how many ways is it possible to distribute material between authors if two people write three chapters each, four people write two chapters each, two people write one chapter each?
Answer: 16!/(26 32 ).
38. 8 third-class chess players participate in a chess tournament, 6 –
second and 2 first-class. Determine the number of such compositions of the first round so that chess players of the same category meet each other (the color of the pieces is not taken into account).
Answer: 420.
39. From the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, all kinds of five-digit numbers are made: those that do not contain identical digits. Determine the number of numbers in
which have the numbers 2, 4 and 5 at the same time.
Answer: 1800.
40. Seven apples and two oranges must be placed in two bags so that each bag contains at least one orange and so that the number of fruits in them is the same. In how many ways can this be done?
Answer: 105.
41. Morse code letters are made up of symbols (dots and dashes). How many letters can you draw if you require that each letter contain no more than five characters?
42. The vehicle trailer number consists of two letters and four numbers.
How many different numbers can you make using 30 letters and 10 numbers?
Answer: 9 106.
43. A gardener must plant 10 trees within three days. In how many ways can he distribute his work over the days if he plants at least one tree a day?
44. From a vase containing 10 red and 4 pink carnations, choose one red and two pink flowers. In how many ways can this be done?
45. Twelve students were given two versions of the test.
In how many ways can students be seated in two rows so that those sitting next to each other do not have the same options, but those sitting next to each other have the same option?
Answer: 2(6!)2.
46. Each of the ten radio operators at point A tries to establish contact with each of the twenty radio operators at point B. As many as possible various options such a connection?
Answer: 2200.
47. Six boxes of different materials are delivered to eight floors of a construction site. In how many ways can materials be distributed among floors? IN
In how many options will no more than two materials be delivered to the eighth floor?
Answer: 86 ; 86 –13 75 .
48. In how many ways can two players be formed into one line? football teams so that two players of the same team do not stand next to each other?
Answer: 2(11!)2.
49. On the bookshelf there are books on mathematics and logic - a total of 20 books.
Show what greatest number options for a set containing 5 books on mathematics and 5 books on logic are possible in the case when the number of books on the shelf for each subject is 10.
Answer: C 5 10–x C 5 10+x(C 5 10) 2.
50 . An elevator carrying 9 passengers can stop on ten floors. Passengers disembark in groups of two, three and four.
In how many ways can this happen?
Answer: 10!/4.
51. “Early in the morning, smiling Igor rushed barefoot to go fishing.”
How many different meaningful sentences can be made using part of the words of this sentence, but without changing their order?
52. In a chess match between two teams of 8 people, the participants in the games and the color of the pieces of each participant are determined by lot. What is the number of different outcomes of the draw?
A 10 6 .
Answer: 28 8!.
53. A and B and 8 other people are standing in line. In how many ways can people be arranged in a queue so that A and B are separated from each other by three persons?
Answer: 6 8! 2!.
54. How many four-digit numbers can be made from the numbers 0, 1, 2, 3, 4, 5,
if a) the numbers are not repeated; b) numbers can be repeated; c) only odd numbers are used and may be repeated; d) should only turn out odd numbers and numbers can be repeated.
Answer: a) 5 5 4 3=300; b) 5 6 = 1080; c) 34; d) 5 6 6 3 = 540.
55. There are 10 subjects studied in the class. In how many ways can you create a schedule for Monday if there are 6 lessons on Monday and all are different?
56. There are m points on one line and n points on a line parallel to it.
How many triangles with vertices at these points can you get?
Answer: mC n 2 nC m 2 .
57. How many five-digit numbers are there that are read the same from right to left and left to right, for example, 67876.
Answer: 9 10 10 = 900.
58. How many different divisors (including 1 and the number itself) does the number have?
35 54 ?
59. In a rectangular matrix A = (a ij )m rows and n columns. Eacha n n = 2n –1.
61. How many four-digit numbers are there in which each subsequent digit is greater than the previous one?
Answer: C 9 4 = 126.
62. How many four-digit numbers are there in which each subsequent digit is less than the previous one?
Answer: C 10 4 = 210.
63. There are p white and q black balls. In how many ways can they be arranged in a row so that no 2
there were no black balls nearby (q p + 1)?
Answer: C q .p 1
64. There are p different books in red bindings and q different books in blue bindings (q p + 1).
In how many ways can they be arranged in a row so that no two blue-bound books stand next to each other?
Answer: C q p! q! .p 1
65. In how many ways can (1, 2, ...n) numbers be ordered so that the numbers 1, 2, 3 are next to each other in ascending order?
Answer: (n – 2)!.
66. There must be 4 speakers at a meeting: A, B, C and D, and B cannot speak before A.
In how many ways can their order be determined?
Answer: 12 = 3! + 2 2 +2.
67. In how many ways can m +n +s objects be distributed into 3 groups, so that one group contains m objects, another -n, and a third -s objects.
Answer: (m + n + s)!.
68. How many integer non-negative solutions does the equation x 1 +x 2 + ... +x m =n have?
Answer: C n .n m 1
69. Find the number of vectors = (1 2 ...n), whose coordinates satisfy the conditions:
1) i (0, 1);
2) i (0, 1, ...k – 1); 3)i (0, 1, ...k i – 1);
4) i (0, 1) and1 +2 + ... +n =r.
Answer: 1) 2n ; 2)k n ; 3)k 1 k 2 ...k n ; 4)
70. What is the number of matrices (a ij), where a ij (0,1) and in which there are m rows and n columns? 1) strings can
repeat; 2) the strings are pairwise different.
Answer: 1) 2m n ; 2) .