Math/Stat 304 counting problems, continued

Note: Carefully show your work. You need not simplify your answers --- particularly when they involve factorials, combinations and/or permutations. The word “subset”, unless otherwise qualified, means “unordered subset.” The word “deck” refers to a standard deck of cards.

1. How many rearrangements are there of the word KOLMOGOROV?

2. 25 students show up at the OZ Fitness & YOGA Center looking for open classes. Only 3 classes are still open: one has 8 spots, one has 11 spots, and one has 6 spots. In how many different ways can the students be arranged in the 3 classes?

3. How many license plates with 4 decimal digits followed by 5 letters do not contain both the number 0 and the letter O?

4. In how many ways can Albertine paint the faces of a regular tetrahedron with four colors if each face is painted a different color? (Assume that two painted tetrahedrons that can be oriented to look the same are considered indistinguishable.)

5. From a standard deck, in how many ways can Swann be dealt a hand of 8 cards containing four-of-a-kind, and two distinct pairs?

6. From a standard deck, in how many ways can Swann be dealt a hand of 7 cards containing exactly three (unspecified) suits?

7. How many different sequences of the numbers {0, 1, 2} of length 10 do not contain any of the subsequences 12, 23, or 31? For example, 3222132111 is such a sequence.

8. A decimal number is called “increasing” if each digit is greater than the previous one (for example, 24579 is one). How many 5 digit increasing numbers are there? 2

9. A 7-digit phone number d1d2d3d4d5d6d7 is called memorable if d1d2d3 is exactly the same sequence as d4d5d6 or d5d6d7 (possibly both). (e.g. 4357435 is memorable). Assuming each di can be any decimal digit (so d1 could be 0), how many memorable telephone numbers are there?

10. How many triangles can be formed with vertices on a 3×3 grid of points?

11. How many 5-letter words have at least one double letter, i.e. two consecutive letters that are the same?

12. A diagonal of polygon is any line segment between vertices which is not an edge of the polygon. How many diagonals does an n-sided polygon have?

13. Albertine lives in a city with a square grid of numbered streets which run east-west and numbered avenues that run north-south. Her house is located on the corner of 0th Street and 0th Avenue. Odette, her aunt, lives at the corner of 5th St. and 3rd Ave. (a) How long is the shortest route (along streets or avenues) to her aunt’s house? How many direct routes can Sally take to her aunt’s house? (b) There is an ATM machine at the corner of 2nd St. and 2nd Ave. If Albertine needs to stop at the store on her way to her Aunt’s, how many direct routes to her Aunt’s house take her through the intersection of 2nd St. and 2nd Ave? (c) At her Aunt’s house Albertine hears on the radio that there has been an accident at the corner of 1st St. and 2nd Ave. Assuming that she avoids this intersection, how many direct routes can Albertine take home? 3