The tutorial for this set of problems can be found here. The answers are at the end of this article.
Questions
1. A bookstore has a selection of 10 books for free. However, any customer is allowed only 3 books. How many ways can a customer choose 3 books?
2. A car race has 8 contestants. A record keeper writes down the car names as they cross the finish line. How many possible ways could the record turn out?
3. Your house has 10 (different) paintings that you are packing. You have 2 burlap bags that can hold 3 paintings and a box that can hold 4 paintings. How many ways can you pack the paintings?
4. You are planning a vacation for 6 days. You can either spend 2 days each at 3 destinations or 3 days each at 2 destinations. There are 12 destinations in your surroundings. How many different vacation plans are there?
5. In a variation of chess, the 16 pieces are randomly arranged in the two back rows of the board. A chess set has 8 pawns, 2 rooks, 2 knights, 2 bishops, a queen, and a king.
- a. What is the probability of getting the standard setup?
- b. What is the probability of getting the pawns all in the front row?
6. Suppose you are playing 5 card poker.
- a. What is the probability of getting a 4 of a kind?
- b. What is the probability of getting a 2 pair (and nothing better)?
7. A fun-size bag of M&M's has 15 M&M's. There are 5 red, 3 green, 3 blue, 2 yellow, 1 brown, and 1 orange.
- a. You randomly take 3 M&M's. What is the probability that all 3 are red?
- b. You randomly take 3 M&M's. What is the probability that 1 is red, 1 is green, and 1 is blue?
- c. You randomly take a non-empty subset of the M&M's. What is the probability that the subset only contains red, green, and blue M&M's?
8. A movie theater has 9 showings in a day. The theater will be playing 3 new movies. There will be 5 showings of one movie, 3 showings of another, and just one showing of the third.
- a. How many ways are there to organize the showings?
- b. Suppose the manager enjoys making completely arbitrary decisions for the showings. What is the probability that no movie is played in consecutive time slots.
