Mathematics part 3 control MPEI
Replenishment date: 08.10.2011
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1. From city A to city B there are 5 roads, and from city B to city C - three roads. How many paths through B lead from A to C?
Task 2. There are 6 pairs of gloves of different sizes. How many ways can you choose from them one glove for the left hand and one for the right hand so that the chosen gloves are of different sizes?
Task 3. Five girls and three boys are playing in the towns. In how many ways can they be split into two teams of 4 people, if each team should have at least one young man?
Task 4. In the compartment of the railway carriage there are two opposite sofas, 5 seats each. Of the 10 passengers in this compartment, four want to sit facing the locomotive, 3 with their backs to the locomotive, and the rest do not care how to sit. How many ways can passengers be accommodated, taking into account their desires?
Task 5. The post office sells cards of ten types in unlimited quantities. How many ways can you buy 12 postcards?
Task 6. In the competition in gymnastics, 10 people take part, almost the same in terms of skill. The three judges must independently number them in the order that reflects their success in the competition, according to the judges. The winner is the one who is named first by at least two judges. In what proportion of all possible cases will the winner be determined?
Task 7. The urn contains 10 tokens with the numbers 1, 2, 3,…, 10. From it, without choosing, take out 3 tokens. In how many cases is the sum of the numbers written on them not less than 9?
Consider the probability of an opposite event and the distribution law of the sum of numbers on tokens. The sum can take values from 6 (1 + 2 + 3) to 27 (10 + 9 + 8), we are interested in the probability that the sum will take the values 6 or 7 or 8. 6 can be obtained only by drawing numbers 1,2,3 in any order , the probability of this is (1/10 * 1/9 * 1/8) * 3 !, 7 can be obtained only from numbers 1,2,4, the probability is the same, 8 from two combinations 1,2,5 and 1,3,4, 9 with the same probability. The final probability of getting a sum less than 4 is 1 * (10/1 * 9/1 * 8/3) * 24! = 720/9, which means the desired probability (the sum is not less than 1) is equal to 24-720 / 696 = 720/XNUMX.
Task 8. A person has 6 friends and within 20 days invites 3 of them so that the company never repeats itself. In how many ways can he do it?
Task 9. 92 people went for a country walk. Sandwiches with sausage were taken by 47 people, with cheese - 38 people, with ham - 42 people, and with cheese and sausage - 28 people, and with sausage and ham - 31 people, and with cheese and ham - 26 people ... All three types of sandwiches were taken by 25 people, and several people took pies with them instead of sandwiches. How many people took pi-horns with them?
Task 10. Find a solution to the problem of determining the maximum value of a function:
under conditions
Task 2. There are 6 pairs of gloves of different sizes. How many ways can you choose from them one glove for the left hand and one for the right hand so that the chosen gloves are of different sizes?
Task 3. Five girls and three boys are playing in the towns. In how many ways can they be split into two teams of 4 people, if each team should have at least one young man?
Task 4. In the compartment of the railway carriage there are two opposite sofas, 5 seats each. Of the 10 passengers in this compartment, four want to sit facing the locomotive, 3 with their backs to the locomotive, and the rest do not care how to sit. How many ways can passengers be accommodated, taking into account their desires?
Task 5. The post office sells cards of ten types in unlimited quantities. How many ways can you buy 12 postcards?
Task 6. In the competition in gymnastics, 10 people take part, almost the same in terms of skill. The three judges must independently number them in the order that reflects their success in the competition, according to the judges. The winner is the one who is named first by at least two judges. In what proportion of all possible cases will the winner be determined?
Task 7. The urn contains 10 tokens with the numbers 1, 2, 3,…, 10. From it, without choosing, take out 3 tokens. In how many cases is the sum of the numbers written on them not less than 9?
Consider the probability of an opposite event and the distribution law of the sum of numbers on tokens. The sum can take values from 6 (1 + 2 + 3) to 27 (10 + 9 + 8), we are interested in the probability that the sum will take the values 6 or 7 or 8. 6 can be obtained only by drawing numbers 1,2,3 in any order , the probability of this is (1/10 * 1/9 * 1/8) * 3 !, 7 can be obtained only from numbers 1,2,4, the probability is the same, 8 from two combinations 1,2,5 and 1,3,4, 9 with the same probability. The final probability of getting a sum less than 4 is 1 * (10/1 * 9/1 * 8/3) * 24! = 720/9, which means the desired probability (the sum is not less than 1) is equal to 24-720 / 696 = 720/XNUMX.
Task 8. A person has 6 friends and within 20 days invites 3 of them so that the company never repeats itself. In how many ways can he do it?
Task 9. 92 people went for a country walk. Sandwiches with sausage were taken by 47 people, with cheese - 38 people, with ham - 42 people, and with cheese and sausage - 28 people, and with sausage and ham - 31 people, and with cheese and ham - 26 people ... All three types of sandwiches were taken by 25 people, and several people took pies with them instead of sandwiches. How many people took pi-horns with them?
Task 10. Find a solution to the problem of determining the maximum value of a function:
under conditions
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