J. Chen, Handout 2, STAT550, 2010
1
STAT550 Applied Probability
http://www-rohan.sdsu.edu/ jchenyp/STAT5502010.htm
1.3. Counting Techniques
The permutations and combinations are very useful tools to compute the proba-
bility of the event. In particular, both methods provide eﬃcient ways to calculate the
number of the sample points in sample space for the equally likely classical probability
model. The terminology of the permutations and combination can be described as:
•
A PERMUTATION is an arrangement of distinct items (objects) in a particular
order
•
A COMBINATION is a selection of items (objects) without considering order
1). Multiplication rule
If procedure (operation)
A
can be done in
m
diﬀerent ways and procedure
B
in
n
diﬀerent ways, the sequence (A,B) can be done in
m
×
n
diﬀerent way.
In general, if procedure
A
i
,
i
= 1
,
···
,k
, can be performed in
n
i
ways,
i
= 1
,
···
,k,
respectively, then the order sequence (
A
1
,A
2
,
···
,A
k
) can be performed in
n
1
×···×
n
k
diﬀerent way.
EX 1.
A traveler may drive any one of three routes from Kansas City to Chicago
and any one of four routes from Chicago to New York City as depicted in Figure.
Solution:
If we count the number of routes from Kansas City to Ne York through
Chicago, we see there are a total of (3)(4) = 12 routes.
EX 2.
The combination lock on a briefcase has two dials, each marked oﬀ with 16
notches. To open the case, a person ﬁrst turns the left dial in a certain direction
for two revolutions and then stops on a particular mark. The right dial is set in a
similar fashion, after having been turned in a certain direction for two revolutions.
How many diﬀerent setting are possible?
Solution:
By using the multiplication rule, opening the briefcase corresponds to
the four-step sequence (
A
1
,A
2
,A
3
,A
4
), then Number of diﬀerent settings =
n
1
.n
2
.n
3
.n
4
=
2
×
16
×
2
×
16 = 1024.
2). Permutations