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Dr.Berg
Chapter 1…Combinatorics
1.1
The Basic Counting Principle
We begin with the counting techniques often used to calculate probabilities.
Theorem
If two tasks
(
experiments, choices, etc.
)
can be done independently in m different
ways for the first and n different ways for the second, then there are mn ways to perform
the tasks.
Proof: Enumerate the outcomes of the first task 1 through
m
, and enumerate the
outcomes of the second task 1 through
n
. Then the combined outcomes can be
enumerated using 2–tuples (
i
,
j
) forming an
m
by
n
array containing
mn
distinct elements.
Note: Mathematical induction generalizes this to any finite number of tasks.
Example A
(2b page 3)
A college planning committee consist of 3 freshmen, 4 sophomores, 5 juniors, and
2 seniors. A subcommittee of 4, consisting of 1 person from each class, is to be chosen.
How many different subcommittees are possible?
Solution
:
We assume that the choices are independent, so there would be
3 4 5 2 =120 possibilities.
Example B
You have 3 pairs of shoes, 3 different colored slacks, and 5 different shirts. You
must choose one of each type for an outfit. If we assume independence, how many
different outfits can be chosen?
Solution
:
There would be
3 3 5 = 45 different outfits.
1.2
Permutations
The different orderings (left to right) of a finite set are called
permutations
of that
set.
These are also called arrangements, rearrangements, orderings, etc.
Proposition
There are
n
!=
n
(
n
 1)(
n
 2)
3 2 1
permutations of n objects. More generally,
there are
n
(
n
 1)
(
n

r
+1) =
n
!
(
n

r
)!
permutations of r objects chosen from among n
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 Spring '08
 Berg
 Combinatorics, Counting

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