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Chapter 4.2 Counting
Read: 4.2
Next Class: 4.3
Motivation
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Section 3.2
We have already seen that different sets can have different
numbers of elements. Even infinite sets can have different
complexities.
Counting of the number of elements in a set (or combinatorics)
is important in statistics where it is used to compute the
probability of the occurrence of a given event.
Combinatorics is an important technique for complexity analysis
of algorithms.
Counting
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Counting
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The idea is to find out how many members are present in a
finite set. Principles of counting answer the following kind
of questions:
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How many four digit numbers can there be if repetition
of numbers are allowed and if repetition of numbers are
not allowed?
If a man has 4 suits, 8 shirts and 5 ties, how many outfits
can he put together?
Counting problems are often cast as a decision problem
where it is analyzed how many possible outcomes exist for a
specific sequence of choices or events.
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Section 3.2
Counting
Multiplication Principle
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Section 3.2
Multiplication Principle: If there are n possible outcomes for a
first event and m possible outcomes for a second event, then
there are nm possible outcomes for the sequence of two events.
Hence, from the multiplication principle, it follows that for two
sets A and B
|AB| = |A|.|B|
Example: A child is allowed to choose one jellybean out of two
jellybeans, one red and one black, and one gummy bear out of
three gummy bears, yellow, green, and white. How many
different sets of candy can the child have?
Counting
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Example: Multiplication Principle
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Section 3.2
There are 23=6 or 32=6 possible outcomes as seen from the
following figures
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Addition Principle
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Section 3.2
Addition Principle: If A and B are disjoint events with n
and m outcomes, respectively, then the total number of
possible outcomes for event “A or B” is n+m.
If A and B are disjoint sets, then |A  B| = |A| + |B| using
the addition principle.
Example:
If you can decide between 5 different cars and 4 trucks, the
total number of choices is 9.
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Addition Principle (cont’d)
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Prove that if A and B are finite sets then
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|A-B| = |A| - |A  B| and |A-B| = |A| - |B| if B  A
(A-B)  (A  B) = (A  B)  (A  B)
= A  (B  B)
=AU
=A
Also, A-B and A  B are disjoint sets, therefore using the addition
principle
|A| = | (A-B)  (A  B) | = |A-B| + |A  B|
Hence, |A-B| = |A| - |A  B|
If B  A, then A  B = B
Hence, |A-B| = |A| - |B|
Section 3.2
Counting
Multiplication and Addition Rules
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Section 3.2
Note: While the addition principle acts within one decision
level, the multiplication principle operates across multiple
decision levels.
Both principles can be combined to address more complex
counting problems.
Example 1: How many possible outcomes are there if you 3
times in a row either roll a die or flip a coin each time?
Example 2: How many different outcomes are there if 3
numbers are drawn in sequence out of a bucket containing the
numbers 0 -9?
Example 3: How many possibilities are there for 3 consecutive
rolls of a die without two consecutive rolls leading to the same
number ?
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Decision trees
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Section 3.2
Trees that provide the number of outcomes of an event
based on a series of possible choices are called decision
trees.
Decision trees are often used to visualize and analyze the
number of possibilities for a selection task.
In a decision tree the nodes correspond to decision points.
The branches leaving such nodes represent different
choices or events. The different levels of the tree thus
correspond to sequential decisions and the leaves represent
the possible outcomes (or elements of the set).
Counting
Decision trees (cont’d)
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Section 3.2
Example: Tony is pitching pennies. Each toss results in heads
(H) or tails (T). How many ways can he toss the coin five times
without having two heads in a row?
There are 13 possible outcomes as seen from the tree.
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Exercises
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Exercise: A dinner allows you to select from five appetizers,
three salads, four entrees, and three beverages.
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1) How many different dinners are there?
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Section 3.2
2) How many different dinners are there if you may have an
appetizer or a salad but not both?
Counting
Exercises
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Exercise: Use the addition and multiplication principles to
determine the number of possible outcomes for the following
problems.
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a) John performs an experiment where he either rolls a die or
flips two coins simultaneously. If he repeats this experiment
4 times in a row, how many possible outcomes can he
obtain?
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Section 3.2
b) If we flip a coin 5 times in a row, how many possible
outcomes are there which contain an odd numbers of heads ?
c) How many different 15-digit binary numbers can be
formed ?
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Exercises
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Section 3.2
Exercise: Draw a decision tree to find the number of binary
strings of length 4 that do not have consecutive 0s.
Counting
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