This document provides examples and definitions for permutations and combinations. It begins by defining a permutation as an arrangement of objects in a specific order where order matters. A combination is a selection of objects without regard to order. The key rules are explained: for permutations, "n objects taken r at a time" is calculated as nPr = n!/(n-r)!, and for combinations "n objects taken r at a time" is calculated as nCr = n!/r!(n-r)!. Several examples are worked through for both permutations and combinations problems to illustrate how to set them up and calculate the number of possible outcomes. Special cases like repeating objects are also addressed.

1. ILLUSTRATING
PERMUTATIONS

2. Warm Up – Find the Mean
and the Standard Deviation.

3. Warm Up – Find the Mean
and Standard Deviation

4. Objective
Find Sample Space
using
Permutations and
Combinations

5. Relevance
 Learn various methods of
finding out how many
possible outcomes of a
probability experiment are
possible.
 Use this information to find
probability.

6. Definition……
 Permutation – an arrangement of
objects in a specific order.
Order Matters!

7. Example……
 How many ways can
you arrange 3
people for a picture?
 Note: You are using
all 3 people
 Answer:
6
1
2
3 



8. Factorial……
 This is the same as
using a factorial:
 Using the previous
example:
1
).....
2
)(
1
(
! 

 n
n
n
n
)
2
3
)(
1
3
(
3
!
3 


6
1
2
3
!
3 




9. Factorial can be found on our
graphing calculator……
 Example: Find 5!
 Calculator Steps:
a. 5
b. Math
c. Prb
d. 4:!
e. Enter

10. Example……
 Suppose a business
owner has a choice of 5
locations in which to
establish her business.
She decides to rank
them from best to least
according to certain
criteria. How many
different ways can she
rank them?
 Answer:
 Note: She ranked ALL
5 locations.
120
1
2
3
4
5
!
5 






11.  What if she only
wanted to rank the
top 3?
 Answer:
 This is no longer a
factorial problem
because you don’t
rank ALL of them.
60
3
4
5 



12. Permutation Rule……
where n = total # of
objects and r = how
many you need.
“n objects taken r at a
time”
)!
(
!
r
n
n
Pr
n



13.  Remember the
business woman
who only wanted to
rank the top 3 out of
5 places?
 This is a
permutation:
60
3
4
5 


60
2
120
!
2
!
5
)!
3
5
(
!
5
3
5 




P

14. Permutations can be found on
the graphing calculator……
 Example: Find
 Calculator Steps:
 5
 Math
 Prb
 2:nPr
 3
 Enter
3
5 P

15. Example……
 A TV news director
wishes to use 3
news stories on the
evening news. She
wants the top 3
news stories out of
8 possible. How
many ways can the
program be set up?
 Answer:
336
3
8 
P

16. Example……
 How many ways can
a chairperson and
an assistant be
selected for a
project if there are 7
scientists available?
 Answer:
42
2
7 
P

17. Example……
 How many different
ways can I arrange
3 box cars selected
from 8 to make a
train?
 Answer:
336
3
8 
P

18. Example……
 How many ways can
4 books be arranged
on a shelf if they
can be selected from
9 books?
 Answer:
3024
4
9 
P

19. A factorial is also a
permutation……
 How many ways can
4 books be arranged
on a shelf?
 You can do 4! or
you can set it up as
a permutation.
Answer:
24
4
4 
P

20. Note……
 0! = 1
and
1! = 1

21. Order Words……
 How many ways can I
 Listen
 Sing
 Read
 1st/2nd/etc
 Pres/Vice-Pres
 Chair/Assistant
 Eat

22. Special Permutation when
letters must repeat……
 Example: How many permutations of
the word seem can be made?
 Since there are 4 letters, the total
possible ways is 4! IF each “e” is
labeled differently. Also, there are 2!
Ways to permute e1e2. But, since they
are indistinguishable, these duplicates
must be eliminated by dividing by 2!.

23.  How many
permutations of the
word seem can be
made?
 Answer:
12
!
2
!
4


24. This leads to another permutation
rule when some things repeat……
 It reads: the # of permutations of n
objects in which k1 are alike, k2 are
alike, etc.
!
!...
!
!
!
3
2
1 p
r
n
k
k
k
k
n
P 

25. Example……
 Find the permutations
of the word
Mississippi.
 Number of Letters
 11 – Total Letters
 1 – M
 4 – I
 4 – S
 2 - P
 Answer:
 You can eliminate the
1!’s because they are
equal to 1.
34650
)
!
2
!
4
!
4
!
1
(
!
11


26. Combinations

27. Definition……
 Combination – a selection of “n”
objects without regard to order.
Order Does NOT Matter!

28. Let’s compare ABCD – Find permutations
of 2 and combinations of 2.
 Permutations of 2:
AB CA
AC CB
AD CD
BA DA
BC DB
BD DC
 Note: AB is NOT the
same as BA.
 Combinations of 2:
AB
AC
AD
BC
BD
CD
 Note: AB is the
same as BA

29.  When different orderings of the
same items are counted
separately, we have a permutation
problem, but when different
orderings of the same items are
not counted separately, we have a
combination problem.

30. Combination Rule……
 Read: “n” objects taken “r” at a time.
!
)!
(
!
r
r
n
n
Cr
n



31. Example……
 How many
combinations of 4
objects are there,
taken 2 at a time?
 Answer:
6
2
12
!
2
!
2
!
4
!
2
)!
2
4
(
!
4
2
4 




C

32. Combinations: There is a key on
the graphing calculator……
 Find
 Calculator Steps:
 4
 Math
 Prb
 3: nCr
 2
 Enter
2
4 C

33. Example……
 To survey opinions
of customers at local
malls, a researcher
decides to select 5
from 12. How many
ways can this be
done?
Why is order is not
important?
 Answer:
792
5
12 
C

34. Example……
 A bike shop owner has
11 mountain bikes in
the showroom. He
wishes to select 5 to
display at a show. How
many ways can a group
of 5 be selected?
Note: He is NOT
interested in a specific
order.
 Answer:
462
5
11 
C

35. Example……
 In a club there are 7
women and 5 men. A
committee of 3 women
and 2 men is to be
chosen. How many
different possibilities are
there?
 The “and” indicates that
you must use the
multiplication rule along
with the combination
rule.
 Answer:
350
10
35
2
5
3
7 


 C
C

36. Example……
 In a club with 7 women and
5 men, select a committee of
5 with at least 3 women.
 This means you have 3
possibilities:
 3W,2M or
 4W,1M or
 5W,0M
 Now you must use
the multiplication
rule as well as the
addition rule.
 The reason for this
is you are using
“and” and “or.”

37. Answer……
 3W,2M:
 4W,1M:
 5W,0M:
Add the totals: 350 + 175 + 21 = 546
350
2
5
3
7 
 C
C
175
1
5
4
7 
 C
C
21
0
5
5
7 
 C
C

38. Example……
 In a club with 7 women and
5 men, select a committee of
5 with at most 2 women.
 This means you have 3
possibilities:
 0W,5M or
 1W,4M or
 2W,3M
 Use the
multiplication rule
and the addition
rule.
 First you multiply,
then you add.

39. Answer……
 0W,5M:
 1W,4M:
 2W,3M:
 Add the totals: 1 + 35 + 210 = 246
1
1
1
5
5
0
7 


 C
C
35
5
7
4
5
1
7 


 C
C
210
10
21
3
5
2
7 


 C
C