# Examination in Mathematics MEI part 3

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1. From the city A to city B are 5 roads in and out of the city in the city with - three roads. How many paths that pass through in the lead from A to C?
Solution: From the city you can go A n

2.Imeetsya 6 pairs of gloves of different sizes. In how many ways you can choose the one glove on his left hand, and one - to the right so that the gloves were selected in different sizes?
Solution: From 6 we get 12 pairs of gloves, calculate how many ways you can select any two of 12. Get the gloves

3.Pyat girls and three boys playing in the towns. In how many ways they can be broken into two teams of 4 people, if each team must have at least one boy?
Solution: Two boys

4. The railway car compartment has two opposite places couch 5 each. Of the 10 passengers on the four compartments are willing to sit face to the engine, 3 - back to the engine, and the rest is indifferent to sit. How many ways can accommodate passengers with regard to their desires?
Solution: So four people sitting face to the engine can accommodate 4 = 24 ways. Three sitting with his back to the engine can be placed 3! = 6 ways. According to the problem, we know that for every sofa can seat 5 people. Then on the couch on which sit four people already

5. The post office selling postcards of ten species in unlimited amounts. In how many ways you can buy 12 cards?
Solution: calculated using the formula:

6. The competition in gymnastics involved nearly 10 people at the same degree of skill. Three judges must be independent of each other to renumber them in order of their success in the competition according to the judges. The winner is the one who first called at least two judges. In what proportion of all possible cases, the winner will be determined?
Solution: So each of the judges put 10 estimates, as all 3 judges will be exposed to 30 estimates. By hypothesis, we know that the winner is the one who first called at least two judges. Then we find out the total number of combinations 30i assessments on two scores. We get all combinations. Number of ratings no ratings winner

7. An urn are 10 tokens with numbers 1, 2, 3, ..., 10. From it, without selecting, removed 3 tokens. In how many cases the amount of numbers written on them not less than 9?
Solution: From an urn with 10 counters can be removed by 3 ways. The following combination of the following three tokens amount less than nine: 1 + 2 + 3 = 6 <9. 1 + 3 + 4 = 8 <9. 1 + 2 + 4 = 7 <9. 1 + 2 + 5 = 8 <9.

8. A man has 6 friends, and within 20 days invites you to the three of them, so that the company has never repeated. In how many ways can he do it?
Solution: So the number "three" from 6 can be formed by the method. The number of ways to invite 20 "threes" is the number of friends

9. In the countryside drove 92 people. Sausage sandwich took 47 people with cheese - 38 people, with ham - 42 people, with cheese and sausage - 28 people, and sausage and ham - 31 people, and with cheese and ham - 26 people. All three types of sandwiches took 25 people, and a few people instead of sandwiches brought with them cakes. How many people have brought the pies?
Solution: On
10. Find the solution of the problem, which consists in determining the maximum value of the function:
under conditions

Solution: Here is the problem to the conical mind what to add in the right part of the additional restrictions under the restrictions unknown
The rank of the system of equations is 3. The rank of the augmented matrix is \u200b\u200balso equal to 3. Therefore, the three variables (base) can be expressed in terms of five variables (free), ie,

By the way, the linear form, or have already expressed through the same free variables.
We have the original table: