Chapter Review

The Multiplication Rule for Counting

1. You are booking a round trip flight for vacation. If there are 4 outbound flight options and 7 return flight options, how many different options do you have?
2. You are putting together a social committee for your club. You’d like broad representation, so you will choose one person from each class. If there are 8 seniors, 12 juniors, 10 sophomores, and 6 first-years, how many committees are possible?
3. The Big Breakfast Platter at Jimbo’s Sausage Haus gives you your choice of 4 flavors of sausage, 5 preparations for eggs, 3 different potato options, and 4 different breads. If you choose one of each, how many different Big Breakfast Platters can be selected?
4. The multiple-choice quiz you’re about to take has 10 questions with 4 choices for each. How many ways are there to fill out the quiz?

Permutations

5. Compute .
6. Compute .
7. Compute .
8. Compute .
9. Compute .
10. Compute .
11. As you plan your day, you see that you have 6 tasks on your to-do list. You’ll only have time for 5 of those. How many schedules are possible for you today?
12. As captain of your intramural softball team, you are responsible for setting the 10-person batting order for the team. If there are 12 people on the team, how many batting orders are possible?

Combinations

13. If you’re trying to decide which 4 of your 12 friends to invite to your apartment for a dinner party, are you using permutations or combinations?
14. If you’re trying to decide which of your guests sits where at your table, are you using permutations or combinations?
15. Compute .
16. Compute .
17. How many ways are there to draw a hand of 8 cards from a deck of 16 cards?
18. In a card game with 4 players and a deck of 12 cards, how many ways are there to deal out the four 3-card hands?

Each of the following exercises involve drawing a Scrabble tile from a bag. These tiles are labeled with a letter and a point value, as follows: A(1), C(3), D(2), E(1), E(1), J(8), K(5), O(1), R(1), R(1).

1. How many ways are there to draw a vowel and then a consonant from the bag?
2. How many ways are there to draw a tile worth an even number of points and then a tile worth an odd number of points from the bag?
3. How many ways are there to draw 4 tiles from the bag without replacement, if order matters?
4. How many ways are there to draw 4 consonants from the bag without replacement, if order matter?
5. How many ways are there to draw 4 tiles from the bag with replacement, if order does not matter?
6. How many ways are there to draw 4 consonants from the bag with replacement, if order does not matter?
7. Give the sample space of the experiment that asks you to draw 2 tiles from the bag with replacement and note their point values, where order doesn’t matter. Give the outcomes as ordered pairs.
8. Give the sample space of the experiment that asks you to draw 2 tiles from the bag without replacement and note their point values, where order doesn’t matter. Give the outcomes as ordered pairs.
9. If you draw a single tile from the bag, what is the probability that it’s an E?
10. If you draw a single tile from the bag, what is the probability that it’s not an A?
11. If you draw 3 tiles from the bag without replacement, what is the probability that they spell RED, in order?
12. If you draw 3 tiles from the bag without replacement, what is the probability that they spell RED, in any order?
13. What are the odds against drawing a vowel?
14. Use your answer to question 12 to find the odds against drawing three tiles without replacement and being able to spell RED.
15. If you draw one tile, what is the probability of drawing a J or a K?
16. If you draw one tile, what is the probability that it’s a vowel or that it’s worth more than 4 points?
17. Suppose you’re about to draw one tile from the bag. Find and .
18. If you draw 2 tiles with replacement, what is the probability of drawing a consonant first and then a vowel?
19. If you draw 2 tiles without replacement, what is the probability of drawing a consonant first and then a vowel?
20. If you draw 10 tiles with replacement, what is the probability that you draw exactly 3 vowels? Round to 3 decimal places.
21. If you draw 100 tiles with replacement, what is the probability that you draw fewer than 35 vowels? Round to 4 decimal places.
22. Find and interpret the expected number of points on the tile, assuming you draw 1 tile from the bag.
23. Find the expected sum of points on 2 tiles, selected without replacement.
24. If your friend offers you a bet where they pay you $10 if you draw a vowel from the bag, but you owe them $5 if you draw a consonant, should you take it? How do you know?

 Attribution

Text Attribution

This text was adapted from Chapter 7 of Contemporary Mathematics, textbooks originally published by OpenStax.