**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?

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

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

**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]

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

**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?

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

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''?

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?