M362K Second Midterm Exam Solutions
March 7, 2002

Problem 1. Political committees

13 Democrats, 12 Republicans and 8 Independents are sitting in a room. 8 of these people will be selected to serve on a special committee.

a) How many different possibilities are there for the committee membership?

You must choose 8 of the 33 people, so there are tex2html_wrap_inline51 possibilities.

b) What is the probability that exactly 5 of the committee members will be Democrats?

This is hypergeometric: tex2html_wrap_inline53

c) What is the probability that the committee will consist of 4 Democrats, 3 Republicans and one independent?

The number of ways to do this is tex2html_wrap_inline55 , so the probability is
tex2html_wrap_inline57 .

Problem 2. Number theory

a) How many solutions exist to the equation x+y+z = 15, where x, y, and z have to be non-negative integers? Simplify your answer as much as possible. [Note: the solution x=12, y=2, z=1 is not the same as x=1, y=2, z=12]

This is the same as the ``give $20 to the mailman, babysitter and garbage man'' problem. We are choosing 15 out of 3 items, repetition OK, order unimportant, and so can do it in tex2html_wrap_inline71 ways.

b) How many solutions exist to the equation tex2html_wrap_inline73 , where x, y and z have to be positive integers?

This is really two copies of part (a). Write tex2html_wrap_inline81tex2html_wrap_inline83tex2html_wrap_inline85 . Then we must have tex2html_wrap_inline87 and tex2html_wrap_inline89 . There are tex2html_wrap_inline91 ways to solve the first equation and tex2html_wrap_inline93 ways to solve the second. The answer is then tex2html_wrap_inline95 .

Problem 3. Spring Break Drinking

On South Padre Island over spring break, 25 underage students try to buy beer with fake IDs. Each has a 10% chance of getting caught, independent of the others.

a) What is the probability that exactly 4 of the students get caught?

This is binomial, with p=0.1 and n=25. The answer is tex2html_wrap_inline101 .

b) What is the probability that 3 or more students get caught?

It's easier to figure the probability that two or fewer get caught, and subtract from 1. The final answer is



c) Evaluate your answer to part (b) numerically.


Problem 4. Grab bag

a) How many 9-letter license plates can be made by rearranging the letters of the phrase ``I HATE MATH''?

There are 2 A's, 2 H's, 2 T's one E, one I, and one M, so there are 9! / (2!2!2!1!1!1!) = 9!/8 ways.

b) 25 children submit science fair projects. The judges will award a 1st-place trophy, a 2nd place trophy, a 3rd place trophy, a 4th place trophy, a 5th place trophy, five identical ``honorable mention'' ribbons (for the 6th-10th best projects), and participation ribbons for the remaining children. In how many different ways can the judges distribute the prizes?

There are 25!/20! to distribute the trophies, then tex2html_wrap_inline111 ways to distribute the honorable mention ribbons, and then one way to distribute the participation ribbons, for a total of 25!/15!5! ways in all.