This document provides instruction on combinations and permutations. It begins with examples of evaluating combinations and permutations, such as finding the number of combinations when choosing 3 items from a group of 8 items (8C3) or the number of permutations when selecting a preferred and substitute color from 12 color options (12P2). It then provides practice problems for students to determine whether a scenario involves combinations or permutations and write the appropriate expression to calculate the number of possibilities. The document aims to teach students the differences between combinations and permutations and how to set up and evaluate expressions to solve combinatorial problems.

1. Lesson 12.6, For use with pages 680-684
Evaluate.
1. 3! 2. 5!

2. Lesson 12.6, For use with pages 680-684
Evaluate.
ANSWER 6
1. 3! 2. 5!
ANSWER 120

3. Combinations
Section 12.6

4. Essential Questions
• What are the differences between
permutations and combinations?
• What are the differences between odds
and probability?
• How is probability used to make
predictions?
• What are the differences between
experimental and theoretical probabilities?

5. • In this section we will learn about selecting
items when order is not important.
• Combination: is a group of items whose
order is NOT important.
• For example – if I were to select three
students from HR to serve on a
committee. The order in which I selected
these 3 does not matter. They form a
committee or group. This is also called a
combination.

6. • Students A, B, C & D (4) are trying out for
a volleyball team. There are slots for only
3 players. What are the combinations of 3
players that could be chosen for the
team?
• Make an organized list.

7. • A-B-C
• A-B-D
• A-C-D
• B-C-D
• There are only 4 combinations of 3 players
chosen from a group of 4 players.

8. EXAMPLE 1 Listing Combinations
County Fair
You have 4 tickets to the county fair and can
take 3 of your friends. You can choose from
Abby (A), Brian (B), Chloe (C), and David (D).
How many different choices of groups of
friends do you have?
SOLUTION
List all possible arrangements of three
friends. Then cross out any duplicate
groupings that represent the same group of
friends.

9. EXAMPLE 1 Listing Combinations
ABC, ACB, BAC,
BCA, CAB, and
CBA all represent
the same group.
ANSWER
You have 4 different choices of groups to take to
the fair.

10. Combination
To find the number of combinations of n
objects taken r at a time.
nCr = nPr
r!
If you selected 4 out of 10
books, the notation would
look like this.
10C4 =10P4
4!

11. Combination
If you selected 4 out of 10 books, the
notation would look like this.
10C4 10P4
4!
= =
10 · 9 · 8 · 7
4 · 3 · 2 · 1
= 210=
5040
24

12. EXAMPLE 2 Evaluating Combinations
Find the number of combinations.
Combination formulaa.
8C3 = 8P3
3!
= 56 Simplify.
3!
=
8 · 7 · 6
=
8 · 7 · 6
3 · 2 · 1
Expand 3! = 3 · 2 · 1.
Divide out common factors
(8 3)!8 P3=
8!
–
= 8 · 7 · 6

13. EXAMPLE 2 Evaluating Combinations
b. 9C 7 =
9P7
7!
Combination formula
9 P7 =
9!
(9 7)!–
= 9·8·7·6·5·4·3
Expand 7!.
= 36 Simplify.
=
9 · 8 · 7 · 6 · 5 · 4 · 3
7!
4
1
=
9 · 8 · 7 · 6 · 5 · 4 · 3
7 · 6 · 5 · 4 · 3 · 2 · 1 Divide out common
factors.

14. GUIDED PRACTICE for Example 2
Find the number of combinations.
2. 8C8 Combination formula
8 · 7 · 6 · 5 · 4 · 3 · 2 · 1=
8!
8 C8 =
(8 - 8)!
8! 8!
0!
=
= 1 Simplify.
8 · 7 · 6 · 5 · 4 · 3 · 2 · 1
=
8 · 7 · 6 · 5 · 4 · 3 · 2 · 1
Expand 8!.= 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1
8p8
8!
=
Divide out common factors.

15. Tell whether the possibilities can be counted using a
permutation or combination. Then write an
expression for the number of possibilities.
3. You want to use a set of 8 lamps for a window
display. Find how many sets you can choose
from 25 lamps in the stock room.
ANSWER combination; 25C8 = 1,081,575
4. How many different ways can you select a
preferred color and a substitute color from a
mail-order catalog offering 12 colors of slacks?
ANSWER permutation; 12P2 = 132

16. Tell whether the possibilities can be counted using a
permutation or combination. Then write an
expression for the number of possibilities.
There are 8 swimmers in the 400 meter freestyle
race. In how many ways can the swimmers
finish first, second, and third?
Swimminga.
SOLUTION
Because the swimmers can finish first, second,
or third, order is important. So the possibilities
can be counted by evaluating 8P3 = 336
a.

17. EXAMPLE 3 Permutations and Combinations
Your track team has 6 runners available for
the 4 person relay event. How many
different 4 person teams can be chosen?
Trackb.
Order is not important in choosing the team
members, so the possibilities can be
counted by evaluating 6C4 = 15.
b.
SOLUTION

18. GUIDED PRACTICE for Example 3
A pizza shop offers 12 different pizza
toppings. How many different 3 - topping
pizzas are possible?
Pizza Toppings7.
Order is not important in choosing the team
members, so the combination possibilities can
be counted by evaluating 12C3 = 220.
SOLUTION

19. GUIDED PRACTICE for Example 3
Student Council
There are 15 members on the student council. In
how many ways can they elect a president and a
vice president for the council?
8.
SOLUTION
Because they elect a president and a vice
president order is important. So the permutation
possibilities can be counted by evaluating
15P2 = 210.

20. Homework
• Page 682 #1-21