, where
x, y and z have to be positive integers?
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?