This document discusses counting theory and introduces permutations and combinations. It defines the fundamental principle of counting as the number of ways independent events can occur by multiplying the number of possibilities for each event. Permutations are arrangements of items where order matters, while combinations are subsets of items where order does not matter. Formulas and examples are provided to calculate permutations and combinations. Guidance is given on determining whether a problem requires permutations or combinations based on whether order is important.

1. 11.6 Counting Theory
Chapter 11 Further Topics in Algebra

2. Concepts and Objectives
 Counting Theory
 Calculate the number of different ways for
independent events to occur (Fundamental Principle
of Counting)
 Identify and calculate permutations
 Identify and calculate combinations
 Be able to identify which method to use with which
problem

3. Counting Theory
 The Fundamental Principle of Counting states that
If n independent events occur, with
m1 ways for event 1 to occur,
m2 ways for event 2 to occur,
… and
mn ways for event n to occur,
then there are
m1 • m2 • … • mn
different ways for all n events to occur.

4. Counting Theory
 Example: As a promotion, a restaurant offered a choice
of 3 appetizers, 7 main dishes, and 4 desserts for $9.99.
How many different 3-course meals are possible?

5. Counting Theory
 Example: As a promotion, a restaurant offered a choice
of 3 appetizers, 7 main dishes, and 4 desserts for $9.99.
How many different 3-course meals are possible?
Each course is an event. The first event can occur in 3
ways, the second event can occur in 7 ways, and the
third event can occur in 4 ways. Therefore, there are
3 • 7 • 4 = 84 possible meals

6. Counting Theory
 Example: A librarian has 5 different books that she
wants to arrange in a row. How many different
arrangements are possible?

7. Counting Theory
 Example: A librarian has 5 different books that she
wants to arrange in a row. How many different
arrangements are possible?
Five events are involved: When we select a book for the
first spot, that leaves 4 choices for the second spot.
Continuing in this fashion gives us 3 choices for the third
spot, 2 choices for the fourth spot, and 1 choice for the
fifth spot:
5 • 4 • 3 • 2 • 1 = 120 different arrangements

8. Permutations
 A permutation of n elements taken r at a time is one of
the arrangements of r elements from a set of n elements.
 Other ways to write Pn, r are and nPr . In Desmos,
use the function nPrn,r.
If Pn, r denotes the number of permutations of
n elements taken r at a time, with r  n, then
 
 


!
,
!
n
P n r
n r
n
r
P

9. Permutations
 Example: Suppose 12 people enter a race. In how many
ways could the gold, silver, and bronze medals be
awarded?

10. Permutations
 Example: Suppose 12 people enter a race. In how many
ways could the gold, silver, and bronze medals be
awarded?
 
 


12!
12,3
12 3 !
P

12!
9!
10 11 12
1320 possibilities

11. Combinations
 A subset of items selected without regard to order is
called a combination.
 Another way to write Cn, r is nCr . As mentioned last
class, in Desmos, we can use nCrn,r.
If Cn, r denotes the number of combinations of
n elements taken r at a time, with r  n, then
 
 
 
 
  
 
!
,
! !
n n
C n r
r n r r

12. Combinations
 Example: How many different committees of 5 people
can be chosen from a group of 9 people?

13. Combinations
 Example: How many different committees of 5 people
can be chosen from a group of 9 people?
 
 
 
 
  
 
9 9!
9,5
5 9 5 !5!
C

9!
4!5!
126 committees

14. How Do I Know Which One?
Permutations Combinations
Number of ways of selecting r items out of n items

15. How Do I Know Which One?
Permutations Combinations
Number of ways of selecting r items out of n items
Repetitions are not allowed

16. How Do I Know Which One?
Permutations Combinations
Number of ways of selecting r items out of n items
Repetitions are not allowed
Order is important Order is not important

17. How Do I Know Which One?
Permutations Combinations
Number of ways of selecting r items out of n items
Repetitions are not allowed
Order is important Order is not important
Arrangements of r items from a set
of n items
Subsets of r items from a set of n
items

18. How Do I Know Which One?
Permutations Combinations
Number of ways of selecting r items out of n items
Repetitions are not allowed
Order is important Order is not important
Arrangements of r items from a set
of n items
Subsets of r items from a set of n
items
 
 


!
,
!
n
P n r
n r
 
 
 
 
  
 
!
,
! !
n n
C n r
n r r
r

19. How Do I Know Which One?
Permutations Combinations
Number of ways of selecting r items out of n items
Repetitions are not allowed
Order is important Order is not important
Arrangements of r items from a set
of n items
Subsets of r items from a set of n
items
Clue words: arrangement,
schedule, order
Clue words: group, committee,
sample, selection
 
 


!
,
!
n
P n r
n r
 
 
 
 
  
 
!
,
! !
n n
C n r
n r r
r

20. How Do I Know Which One?
 Caution: Not all counting problems lend themselves to
either permutations or combinations. Whenever a tree
diagram or the fundamental principle of counting can be
used directly, use it.

21. Classwork
 11.6 Assignment (College Algebra)
 Page 1049: 2-20 (even); page 1034: 24-44 (4); page
1024: 42-48 (even)
 11.6 Classwork Check
 Quiz 11.4