University of California, Los Angeles
Department of Statistics
Statistics 100A
Instructor: Nicolas Christou
Combinatorial analysis
Basic principle of counting:
Suppose two experiments are to be performed. Then, if the first experiment can result in
m
outcomes and if for each outcome of the first experiment there are
n
outcomes of the second
experiment, then all together there are
m
×
n
possible outcomes.
Examples:
Permutations:
How many different
ordered
arrangements of the letters
A, B, C
are possible?
There are
are 6
permutations
:
ABC, ACB, BAC, BCA, CAB, CBA
. Or, using the basic principle of
counting we can find the number of permutations as follows: 3
×
2
×
1 = 6.
In general
n
objects can be ordered in
n
×
(
n

1)
×
(
n

2)
× · · · ×
1 =
n
! ways.
Each
arrangement it is called a
permutation
.
Example: In how many ways can 4 math books, 3 chemistry books, and 2 history books can
be ordered so that books of the subject are together?
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Suppose
k
objects are to be selected and ordered from
n
objects (
k < n
).
Say,
n
= 4 (
A, B, C, D
), and
k
= 3. Let’s list all the possible permutations:
ABC
BCD
CDA
DAB
ABD
BCA
CDB
DAC
ACB
BDA
CAB
DBC
ACD
BDC
CAD
DBA
ADB
BAC
CBD
DCA
ADC
BAD
CBA
DCB
As we observe there are 24 permutations. Much easier, we can find the number of permuta
tions using the basic principle of counting as follows: 4
×
3
×
2 = 24.
In general, the number of ways that
k
objects can be selected and ordered from
n
objects
are:
n
×
(
n

1)
×
(
n

2)
× · · · ×
(
n

k
+ 1). This can be simplified if we multiply and
divide by (
n

k
)!:
n
×
(
n

1)
×
(
n

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 Fall '07
 Wu
 Statistics, Counting, Playing card, basic principle

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