Problems 1-5 are worth 20 points each. Problem 6 is worth 15 bonus quiz points. Show all your work and explain your reasoning. An answer by itself is worth at most half credit.

1. Let A, B U be sets with n(U) = 88, n(A)=16, n(B)=40 and n(A B) = 50.
   1. Find n(A B).
   2. Find n(A \ B). ( note: A \ B = A B' ).
   3. Find n(A' B' ).
   4. Find n(A (A B)).
   5. Explain why it is impossible to computer n( (A A ) B) from the information given.
   
   

2. A survey of 100 students finds that eighteen are accounting majors, forty are in-state students, and eight are out-of-state accounting majors. Use a Venn diagram to determine how many in-state students are not accounting majors.
   

3. Using only odd digits, how many four digit numbers are there in which each digit used occurs exactly twice? How many are there in which each digit used occurs more than once?
   

4. Two fair dice are rolled, one red and one green.
   1. Define an equiprobable sample space for this experiment.
   2. Find the probability that the two numbers obtained differ by at least two (e.g. a four and a six would be ok, but not a four and a five.) Express your answer as a fraction in lowest terms.
   
   

5. Before the system changes of the last ten years or so, a three digit (geographic) telephone are code had to satisfy the following three conditions:
   1. The first digit was neither a zero nor a one;
   2. The second digit was either a zero or a one, and;
   3. The last two digits were not the same.
   How many different area codes were possible under this system?
   

6. Bonus

   Use a tree diagram to determine how many arrangements (permutations) there are of the digits 1,2,3 and 4 in which no digit is in its natural position; that is 1 is not the first; 2 is not the second, etc. (There are some "blind alleys" in this tree diagram. For example, you could begin 3 1 2 and they you could not complete the process without putting a 4 in the fourth space. Chose a way of marking these so that you do not count them.)