Download Counting problems: permutations versus combinations

Name:
Date:
Algebra 2
Counting Review
Objective: You will correctly solve varied review problems involving counting.
Review problems
1. a. Based on the symmetry relationship in Pascal’s Triangle, what other combination number
has the same value as 9C5?
b. Based on the adding relationship in Pascal’s Triangle, what are the two combination
numbers that can be added to get 9C5?
c. Show how to find the value of 9C5 using arithmetic (not using a calculator shortcut, and
not using Pascal’s Triangle). Show your work.
d. Find the value of 9C5 again, by writing Pascal’s Triangle from the beginning through the
row that contains 9C5.
e. Write a counting problem of your own, whose answer is given by 9C5.
Name:
Date:
Algebra 2
2. a. Identify another combination number that is equal to
b. What single combination number is equal to
17C5
12C7
.
+ 17C6 ?
c. What is the easiest way to find the sum
10C0 + 10C1 + 10C2 + 10C3 + 10C4 + 10C5 + 10C6 + 10C7 + 10C8 + 10C9 + 10C10 ?
Directions for problems: First write the calculation that needs to be done, then write the answer.
Using calculator shortcuts is allowed, but you still must show what calculation you are doing. If
you only write the final answer, you will not get full credit. For example, first write
5 · 4, then write 20. Or, first write 5P2, then write 20. Don’t just write 20.
3. Out of these 6 school subjects
English, Social Studies, Math, Science, Foreign Language, Physical Education
you are asked to choose your 1st favorite, 2nd favorite, and 3rd favorite subjects.
How many different ways could you make these choices?
4. There are 36 students on the high school Senate. Suppose that two of these students must be
elected to the positions of Moderator and Secretary. How many different ways could the
election turn out?
5. There are 60 homerooms in the school. Suppose that 5 lucky homerooms will be randomly
chosen to receive pizza parties. How many different ways could the pizza parties be awarded?
6. The letters APRIL can be rearranged in various ways: ARILP, RPLIA, LIPAR, etc.
How many different ways can this be done?
Name:
Date:
Algebra 2
7. A push-button lock has five buttons that look like this:
[1] [2] [3] [4] [5]
A lock combination is a sequence of six button presses (examples: 231154; 315232; 444444).
How many different lock combinations are possible?
8. Suppose you are ordering a salad. You have the choice of whether or not to include each of
these items in your salad:
tomato, cucumber, carrot, onion, radish, croutons, dressing
How many different ways could you order your salad?
9. There are 5 multiple-choice questions with choices (A), (B), and (C). How many different
ways could this set of questions be answered?
10. An ATM (banking machine) requires a 4-digit PIN (password number). The PIN can be any
four digits (examples: 0123, 5555). How many different PIN’s are there?
11. Suppose that next week, it rains on 3 out of the 7 days. How many different possibilities exist
for the set of days that it rains?
Name:
Date:
Algebra 2
12. From 12 available student-athletes, a track coach must pick 4 different people to compete in
the long jump, the high jump, the javelin throw, and the hurdles. How many different ways
could the coach make this set of choices?
13. You flip a coin 7 times. How many possible coin flip sequences have exactly 2 heads and 5
tails (example: HTTTTHT)?
14. Suppose you are ordering a salad. You have the choice of whether or not to include each of
these 7 toppings in your salad:
tomato, cucumber, carrot, onion, radish, croutons, dressing
a. How many different ways could you order your salad?
b. How many possible salads have exactly 4 toppings?
15. TGI Friday’s Restaurant has a special deal to get an appetizer, entrée and dessert all for one
price. Below are the options for each course of the meal.
Appetizer
Entrée
Dessert
Classic Wedge Salad
Parmesan-Crusted Chicken
Chocolate Peanut Butter Pie
Fried Mozzarella
Chicken Fingers
Vanilla Bean Cheesecake
Tuscan Spinach Dip
Cheeseburger
Pot Stickers
Brushetta Chicken Pasta
Shrimp Key West
Petite Sirloin
How many different ways could you order this deal?
Name:
Date:
Algebra 2
Answers to today’s review problems
1. a. 9C4
b. 8C4 + 8C5 c. 9C5 =
9 87 6 5
= 126
1 2  3  4  5
d. Triangle should include this row. The underlined value is 9C5.
1, 9, 36, 84, 126, 126, 84, 36, 9, 1
e. Any problem involving choosing 5 out of 9 things without an order.
2. a.
12C5
b.
3.
6P3
= 120
4.
36P2
= 1260
5.
60C5
= 5,461,512
6. 5! = 120
7. 56 = 15625
8. 27 = 128
9. 35
10. 104
11. 7C3
12.
12P4
13. 7C2
14. a. 27 b. 7C4
15. 4 * 6 * 2
18C6
c. 210