CMPT 285 Discrete Mathematics

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CMPT 285Discrete Mathematics

• by Kossi Edoh
• Department of Computer Science
• Montclair State University

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Section 4.1 The basics of Counting
The Rule of Sum If A and B are disjoint sets the
A?B AB Example Suppose statement
labels in a programming must be a single
letter or a single decimal digit. since
a label cannot be both at the same time, the
number of labels of letters of
decimals 26
10 The Rule of Products A x B
A B Example Statement labels in Basic can
be either - a single letter or -
a letter followed by a digit
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Example Continued

• Find the number of possible labels.
• Solution
• The set of all labels L S ? D has the
  partitions
• the set of single letter labels S
• the set of single letters followed by a digit
  D
• The elements of D are ordered pairs of the form
  a,d
• where a is an alphabetic character and d is a
  digit.
• By the rule of product the cardinality of D is
  the product
• of the cardinality of the sets
• (the alphabetic characters)(the decimal
  digits)(26)(10)
• L D S 26 260 286

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The Principle of Inclusion-Exclusion
If A and B are not disjoint A?? B
A B - A ?B Example Count the number of
bits strings of length 4 which begin with a 1 or
end with a 00 Solution A set of bit strings
which begin with 1 1000, 1001, 1010, 1011, 1100,
1101, 1110, 1111 B set of bit
strings which end with 00 0000, 0100, 1000, 1100
A?? B 8 4 2 10
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Example
Example A class contains 25 computer science
majors, 13 math majors and 8 with joint majors
in math and computer science. How many students
are in the class? C ? M C M - C ?
M 25 13 - 8
30
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Example cont

• A ? B ? C A B C - A ? B - B ? C
  - A ? C
• A ? B ? C
• Example
• 1232 students have taken Spanish
• students have taken French
• students have taken Russian
• 103 students have takes Spanish and French
• 23 students have taken Spanish and
  Russian
• 14 students have taken French and
  Russian
• 2092 have taken French, Russian and Spanish
• 2092 1232 879 114 - 103 - 23 - 14 A ? B
  ? C
• Therefore A ? B ? C 7

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Section 4.3 Permutation and Combination
Given a set of n objects in an urn We are going
to pick (select) r objects from the urn in a
sequence. After we choose an object we
can replace it (selection with replacement)
or not (selection without replacement) Permut
ation selection without replacement of r objects
from the urn with n objects. (order
matters) choose the first object in n ways
choose the second object in n-1 ways
choose the rth
object in n-r1 ways
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Permutation and Combination continued
The number of permutations of n things taken r at
time P(n,r) n (n-1) (n-2) (n-r1)
n!/(n-r)! Example Count the number of
injections from the set A to the set B, where
A ? B Solution There are P(B, A)
injections Combination is a selection without
replacement but order does not matter It is
equivalent to selecting subsets of size r from a
set of size n.
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Combination
The number of combinations of n things taken r at
a time n choose r
the binomial coefficient C(n,r)
(n,r) P(n,r)/ P(r,r)
n! / ((n-r)! r!) Example How many subsets
of size r can be constructed from a set of n
objects? Solution C(n,r)
n Corollary ?C(n,r) 2n r0
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Example
Suppose you flip a fair coin n times. How many
different ways can you get no head
C(n,0) exactly one
head C(n,1) exactly two heads
C(n,2) exactly r heads
C(n,r) at least 2 heads
2n C(n,0) - C(n,1) Pascals Identity
C(n1,k) C(n, k-1) C(n,k)
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Example continued
How many bit strings of length 4 have exactly 2
ones ( exactly two zeros) Solution C(4,2)
4!/ 2! 2! 6 How
many bit strings of length 4 have at least 2
ones Solution exactly 0 one exactly 1 one
exactly 2 ones C(4,0) C(4,1) C4,2) 1 4
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