Introduction
Permutations
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics  Permutations
232
What is Combinatorics
Combinatorics, the study of arrangements of objects, is an
important part of discrete mathematics.
This subject was studied as
long ago as the seventeenth century, when combinatorial questions
arose in the study of gambling games.
Enumeration is the counting of objects with certain properties
Combinatorics is used in
 Discrete probability:
What is the probability to guess a 6symbols
password in the first attempt?
 Analysis of algorithms:
Why a comparison sort algorithm cannot
be more efficient than
O(n log n)?
 Probabilistic proofs:
Show that the local search algorithm with
high probability does not find a good solution to a problem.
Discrete Mathematics  Permutations
233
The Rule of Sum
If a first task can be performed in
m
ways,
while a second task
can be performed in
n
ways,
and the two tasks cannot be
performed simultaneously, then performing either task can be
accomplished in any one of
m + n
ways.
Example:
Suppose that either a member of the mathematics faculty
or a student who is a mathematics major is chosen as a
representative to a university committee.
How many different
choices are there for this representative if there are 37 members of
the mathematics faculty and 83 mathematics majors.
Solution:
There are 37 ways to choose a faculty member, and
there are 83 ways to choose a student.
Choosing a faculty member
is never the same as choosing a student.
By the rule of sum there are
37 + 83 = 120
possible choices.
Discrete Mathematics  Permutations
234
The Rule of Product
If a procedure can be broken down into two stages, and if there are
m
possible outcomes of the first stage and if,
for each of these
outcomes, there are
n
possible outcomes for the second stage,
then the total procedure can be carried out, in the designated order,
in
m
⋅
n
ways.
Example:
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 AndreiBulatov
 Combinatorics, Sanchez, faculty member, KNR

Click to edit the document details