COMBINATORIAL MATHEMATICS time limit test

Test #1

• Choose and submit not more than 6 problems. • Write the solutions of your chosen problems in order. • The maximum grade for this test is 30 points. • Simplify as much as possible your answers. • No points are given to answers without the supporting work. • Write clearly all the steps leading to your answers.

PROBLEMS:

1. (5 points) How many integers less than 3500 have the following properties:

(a) The digit 2 is not used. (b) All digits are distinct.

2. (5 points) In how many ways 12 people can be seated at a round table such that A is always sited next to B and C always sited next to D?

3. (5 points) How many sets of three integers between 1 and 20 are possible if no two consecutive integers are to be in a set?

4. (5 points) In how many ways can three red and four blue rooks be placed on a 9 × 9 board so that no two rooks can attack each other?

5. (5 points) Determine the number of 7-permutations of the multiset S = {3 · a, 2 · b, 3 · c}.

6. (5 points) There are 16 identical chairs aligned in a row. In how many ways one can choose 5 of them such that no two chairs are next to each other?

7. (5 points) In how many ways one can put 50 books on 5 shelves if the first two shelves have at least 5 books each and the last three shelves have at least 3 books each?

8. (5 points) ) A 5-member committee are chosen from a group of 8 men and 10 women. How many different committees can be made if the committee consists of at least 3 men, and one particular man and one particular woman do not wish to be on the committee together?

9. (5 points) How many 6-letter words can be constructed using 6 of the 7 letters in “ALAKAKA”?

10. (5 points) Show that if 6 distinct integers are chosen from the set {1,2,...,15}, then there are always two which differ by at most 2.