10-7 Combinations
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
10-7 Combinations
Warm Up
Decide whether each event is independent or
dependent. Explain your answer.
1. Bill picks a king from a pile of cards and keeps it.
On his next turn he tries for a queen.
Dependent; there are fewer cards from which to
choose.
2. the chance of it raining on the third Tuesday of the
month after it has rained on the first Tuesday
Independent; the rain on the first Tuesday has no
effect on the weather two weeks later.
10-7 Combinations
Problem of the Day
There are 7 players. Can more teams of 2
different people or 5 different people be
formed? (Hint: Look for a shortcut.)
The same number of both can be formed. For
every 5-person team there is a 2-person team
(the 2 players left out).
10-7 Combinations
I can find the number of possible
combinations.
10-7 Combinations
Vocabulary
combination
10-7 Combinations
Mrs. Logan’s class has to read any two of the following
books.
1. The Adventures of Tom Sawyer, by Mark Twain
2. The Call of the Wild, by Jack London
3. A Christmas Carol, by Charles Dickens
4. Treasure Island, by Robert Louis Stevenson
5. Tuck Everlasting, by Natalie Babbit
How many possible combinations of books could the
students choose?
10-7 Combinations
A combination is a grouping of objects
or events in which the order does not
matter. For example, a student can
choose books 1 and 2 or books 2 and 1.
Since the order does not matter, the
two arrangements represent the same
combination. One way to find possible
combinations is to make a table.
10-7 Combinations
Additional Example 1: Using a Table to Find
Combinations
Kristy’s Diner offers customers a choice of 4 side
dishes with each order: carrots, corn, french
fries, and mashed potatoes. How many different
combinations of 3 side dishes can Kareem
choose?
Begin by making a table showing all the possible
choices of side dishes taken three at a time.
1, 2, 3
2, 1, 3
3, 1, 2
4, 1, 2
1, 2, 4
2, 1, 4
3, 1, 4
4, 1, 3
1, 3, 4
2, 3, 4
3, 2, 4
4, 2, 3
10-7 Combinations
Additional Example 1 Continued
Because order does not matter, you can eliminate
repeated triples. For example 1, 2, 3 is already
listed, so 2, 1, 3 can be eliminated.
1, 2, 3
2, 1, 3
3, 1, 2
4, 1, 2
1, 2, 4
2, 1, 4
3, 1, 4
4, 1, 3
1, 3, 4
2, 3, 4
3, 2, 4
4, 2, 3
There are 4 possible combinations of 3 side dishes
Kareem can choose with his order.
10-7 Combinations
Check It Out: Example 1
Jim is packing for a trip. He has 4 different
colors of shirts: red, green, blue, and yellow
(r, g, b, y). He only has room for 3 of them.
How many different combinations of 3 shirts
can Jim choose for his trip?
Begin by making a table of all the possible
choices of shirts taken three at a time.
r, g, b
g, r, b
b, r, g
y, r, g
r, g, y
g, r, y
b, r, y
y, r, b
r, b, y
g, b, y
b, g, y
y, g, b
10-7 Combinations
Check It Out: Example 1 Continued
Because order does not matter, you can eliminate
repeated triples. For example r, g, b is already listed
so g, r, b can be eliminated.
r, g, b
g, r, b
b, r, g
y, r, g
r, g, y
g, r, y
b, r, y
y, r, b
r, b, y
g, b, y
b, g, y
y, g, b
There are 4 possible combinations of 3 shirts
that Jim can choose to bring on his trip.
10-7 Combinations
Additional Example 2: Problem Solving Application
Lara is going to make a double-dip cone
from a choice of vanilla, chocolate, and
strawberry. She wants each dip to be a
different flavor. How many different cone
combinations can she choose from?
1
Understand the Problem
Rewrite the question as a statement.
• Find the number of possible combinations of
two flavors Lara can choose.
List the important information:
• There are three flavor choices in all.
10-7 Combinations
Additional Example 2 Continued
2
Make a Plan
You can make a tree diagram to show the
possible combinations.
10-7 Combinations
Additional Example 2 Continued
3
Solve
Strawberry
Vanilla
Vanilla
Chocolate
Chocolate
Strawberry
Chocolate
Strawberry
Vanilla
The tree diagram shows 6 possible ways to combine
two flavors, but each combination is listed twice. So
there are 6 ÷ 2 = 3 possible combinations.
10-7 Combinations
Additional Example 2 Continued
4
Look Back
You can also check by making a table. The vanilla
can be paired with two other flavors and the
chocolate with one. The total number of possible
pairs is 2 + 1 = 3.
10-7 Combinations
Check It Out: Example 2
Hamilton had three types of cookies:
sugar, chocolate chip, and oatmeal. He
wants to put them into packages, with
each package containing two different
flavors of cookies. How many different
packages can he make?
1
Understand the Problem
Rewrite the question as a statement.
• Find the number of possible combinations that
Hamilton can choose.
List the important information:
• There are three cookie types in all.
10-7 Combinations
Check It Out: Example 2 Continued
2
Make a Plan
You can make a tree diagram to show the
possible combinations.
10-7 Combinations
Check It Out: Example 2 Continued
3
Solve
Sugar
Oatmeal
Chocolate chip
Sugar
Chocolate chip
Oatmeal
Oatmeal
Chocolate chip
Sugar
The tree diagram shows 6 possible ways to combine
two types, but each combination is listed twice. So
there are 6 ÷ 2 = 3 possible combinations.
10-7 Combinations
Check It Out: Example 2 Continued
4
Look Back
You can also check by making a table. The oatmeal
can be combined with two other types and the
chocolate chip with one. The total number of possible
pairs is 2 + 1 = 3.
10-7 Combinations
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
10-7 Combinations
Lesson Quiz
1. A pizzeria has on special a pizza with 2 toppings
for $9.95. It offers pepperoni, olives, tomato,
sausage, bacon, mushrooms, and ham as
toppings. How many different specials could you
build? 21
2. Jaime has a red shirt and a blue shirt. She has a
green scarf, a yellow scarf, and a purple scarf.
How many different shirt-scarf combinations can
she wear? 6
3. Karl has to pick two reptiles to write a report on.
If there are 8 possible reptiles to choose from,
how many pairs are possible? 28
10-7 Combinations
Lesson Quiz for Student Response Systems
1. An ice cream parlor offers a discount on a triple
sundae that is made of 3 flavors. You can choose
from chocolate, vanilla, strawberry, mint, and
butterscotch. How many different combinations are
possible?
A. 5
B. 6
C. 8
D. 10
10-7 Combinations
Lesson Quiz for Student Response Systems
2. On the opening day, a restaurant offers free
drinks and desserts to all the customers. A
customer can choose from apple juice, orange
juice, grape juice, and cranberry juice. He can
choose from ice cream, cake, and pudding as
desserts. How many different juice-dessert
combinations are possible?
A. 7
B. 12
C. 15
D. 24
10-7 Combinations
Lesson Quiz for Student Response Systems
3. How many 2-letter combinations are possible
from the letters in the word MATH?
A. 2
B. 4
C. 6
D. 8