Problems 1-4 are worth 25 points each. Except for

, part b), you must show all your work to receive more than half credit.

1. The sets A and B are not disjoint and neither is a subset of the other. .
   1. Find the largest and smallest possible value of .
   2. Find the largest and smallest possible value of . [Hint: you may wish to use a Venn diagram; or the formula: may be helpful.]
   
   

2. The twelve-member board of directors of a small company appoints 3 of its members to a task force. You have two friends on the board.
   1. How many different task-forces are possible? (Order does not matter.)
   2. How many ways could the task force be chosen to include neither of your friends?
   3. Because of political disagreements which could not be settled by reasonable discussion, the board finally decided to choose the task force randomly from its membership. What is the probability that at least one of your friends ends up on the task force?
   
   

3. Two fair dice are rolled, one red and one green.
   1. Set up and describe explicitly the thirty-six element sample space for this experiment.

   2. Let A denote the event that at least one of the dice yields an odd number; let B denote the event that both dice yields odd numbers, and let D denote the event that one die yields an odd number and the other an even number. State whether each of the following is true or false:
      1. 
      2. 
      3. 
      4. 

   3. Find the probability of each of the events from part b).

   
   

4. A series of games is played between two players. The champion, A, must win two games to win the series. The challenger, B, must win three games to win the series. How many sequences of outcomes of games are possible? (Use a tree diagram.)
   

5. How many five-digit numbers are there in which the only digits which appear are , and no digit appears more than twice?