A

You may use one sheet of handwritten notes, but no other sources. The exam consists of five questions, and lasts fifty minutes. Please work all five problems. Please show all work clearly and in reasonable detail. Answers without appropriate supporting work or requested explanations may not receive full credit. No calculators are allowed.

1. (20 points. Each part is worth 10 points.)
   1. Show that among any group of five (not necessarily consecutive) integers, there are two with the same remainder when divided by 4.
   2. What is the probability that a positive integer not exceeding 100 selected at random is divisible by 5 or 7? (Each integer is equally likely to be picked.)
2. (10 points) Seven people are to be seated around a circular table. The table has no ends, so it doesn’t matter who sits in which chair. But it does matter how the people are seated relative to each other. In other words, two seatings are considered the same if one is a rotation of the other. How many different ways can the people be seated? (Express your answer in factorials and/or powers of integers.)
3. (15 points) Five cards are dealt from a standard deck of 52 cards. What is the probability that the hand contains two pairs, that is, two cards of one denomination, two cards of another denomination, and a fifth card of a third denomination? (Express your answer in factorials and/or powers of integers.)
4. (20 points)
   1. (15 points) Let r and n be integers ≥ 2. Prove that rⁿ = ∑ _k=0ⁿ (r - 1)^k.
   2. (5 points) Illustrate the result of part 4a by expressing 5³ as a sum of powers of 4.
5. (35 points. Each part is worth seven points.)

   A person giving a party wants to set out 15 assorted cans of soft drinks for her guests. She stops at a store that sells five different types of soft drinks.

   (Express your answer to each part in factorials and/or powers of integers.)

   1. How many different selections of cans of 15 soft drinks can be made?
   2. How many different selections of cans of 15 soft drinks can be made which have at least two cans of each of the five types?
   3. If the different kinds of soft drinks are equally likely, what is the probability that a selection contains at least two cans of each type?
   4. Now assume there are only twelve cans of ginger ale available. How many different selections of cans of 15 soft drinks can be made?
   5. The 15 cans purchased comprised 6 cans of root beer, 4 cans of ginger ale, 2 cans of cola, 2 cans of orange soda, and one can of blueberry soda. The cans are placed in a single line on a counter. In how many ways can they be arranged?