Counting Techniques
MA30204, Introduction to probrabilities
Sittichoke Som-am
October 24, 2012
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Fundamental Principle of Counting
1
Addition Principle
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Fundamental Principle of Counting
1
Addition Principle
2
Multiplication Principle
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Fundamental Principle of Counting
1
Addition Principle
2
Multiplication Principle
Consider a 4-element set A = {a, b, c, d}. In how many ways can
we form a 2-element subset of A?
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Fundamental Principle of Counting
1
Addition Principle
2
Multiplication Principle
Consider a 4-element set A = {a, b, c, d}. In how many ways can
we form a 2-element subset of A?
Example
A group of students consists of 4 boys and 3 girls. How many ways
are there to select 2 students of the same sex from the group?
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Addition principle
Theorem (The Addition Principle)
Suppose that there are n1 ways for the event E1 to occur and n2
ways for the event E2 to occur. If all these ways are distinct, then
the number of ways for E1 or E2 to occur is n1 + n2 .
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Addition principle
Theorem (The Addition Principle)
Suppose that there are n1 ways for the event E1 to occur and n2
ways for the event E2 to occur. If all these ways are distinct, then
the number of ways for E1 or E2 to occur is n1 + n2 .
Theorem
If A1 , A2 , ..., An , n > 2, are finite sets which are pairwise disjoint,
i.e. Ai ∩ Aj = ø for all i, j with 1 < i < j < n, then
|A1 ∪ A2 ∪ A3 ∪ . . . An | = |A1 | + |A2 | + |A3 | + . . . + |An |
or, in a more concise form:
n
n
[
X
|Ai |
Ai =
i=1
Sittichoke Som-am
i=1
Counting Techniques MA30204, Introduction to probrabilities
Example
Example
A From town X to town Y, one can travel by air, land or sea.
There are 3 different ways by air, A different ways by land and 2
different ways by sea as shown in Figure 1
Figure: Figure 1
How many ways are there from X to Y?
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Example
Example
Find the number of squares contained in the Ax A array (where
each cell is a square) of Figure 2
Figure: Figure 2
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Example
Example
Find the number of routes from X to Y in the one-way system
shown in Figure 3
Figure: Figure 3
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
The Multiplication Principle
Mr. Tan is now in town X and ready to leave for town Z by car.
But before he can reach town Z, he has to pass through town Y.
There are 4 roads (labeled 1, 2, 3, 4) linking X and Y, and 3 roads
(labeled as a, b, c) linking Y and Z as shown in Figure 2.1. How
many ways are there for him to drive from X to Z?
Figure: Figure 2.1
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
The Multiplication Principle
Theorem (The Multiplication Principle)
Suppose that an event E can be split into two events E1 and E2 in
ordered stages. If there are n1 ways for E1 to occur and n2 ways
for E2 to occur, then the number of ways for the event E to occur
is n1 n2 .
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
The Multiplication Principle
Theorem (The Multiplication Principle)
Suppose that an event E can be split into two events E1 and E2 in
ordered stages. If there are n1 ways for E1 to occur and n2 ways
for E2 to occur, then the number of ways for the event E to occur
is n1 n2 .
Example
How many ways are there to select 2 students of different sex from
a group of 4 boys and 3 girls?
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
The Multiplication Principle
Theorem (The Multiplication Principle)
Suppose that an event E can be split into k events E1 , E2 , · · · , Ek
in ordered stages. If there are n1 ways for E1 to occur, n2 ways for
E2 to occur,..., and nk ways for Ek to occur, then the number of
ways for the event E to occur is given by n1 n2 · · · nk
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
The Multiplication Principle
Theorem (The Multiplication Principle)
Suppose that an event E can be split into k events E1 , E2 , · · · , Ek
in ordered stages. If there are n1 ways for E1 to occur, n2 ways for
E2 to occur,..., and nk ways for Ek to occur, then the number of
ways for the event E to occur is given by n1 n2 · · · nk
Example
There are four 2-digit binary sequences: 00,01,10,11. There are
eight 3-digit binary sequences: 000,001,010,011,100,101, 110, 111.
How many 6-digit binary sequences can we form?
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Example
Example
Figure 2.4 shows 9 fixed points a, b, c, · · · , i which are located on
the sides of ∆ABC . If we select one such point from each side and
join the selected points to form a triangle, how many such
triangles can be formed?
Figure: Figure 2.4
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Example
Example
Find the number of triangles that can be formed using the 9 fixed
points of Figure 2.4 as vertices.
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Application
Proposition
Let n be a positive integer, and let n = p1a1 p2a2 · · · pkak be a prime
decomposition of n. Then n has
(a1 + 1)(a2 + 1) · · · (ak + 1)
positive integer divisors, including 1 and itself
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
Application
Proposition
Let n be a positive integer, and let n = p1a1 p2a2 · · · pkak be a prime
decomposition of n. Then n has
(a1 + 1)(a2 + 1) · · · (ak + 1)
positive integer divisors, including 1 and itself
Proposition
Let n be a positive integer, and let n = p1a1 p2a2 · · · pkak be a prime
decomposition of n. Then n has
(2a1 + 1)(2a2 + 1) · · · (2ak + 1)
distinct pairs of ordered positive integers (a, b) such that their
least common multiple is equal to n.
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities
References
[1] Titu Andresscu and Zuming Feng, A path to combinatorics for
undergraduates Counting Strategies
[2]
Koh Khee Meng and Tay Eng Guan, Counting,
World Scientific Publishing Co. Pte. Ltd.
2002.
Sittichoke Som-am
Counting Techniques MA30204, Introduction to probrabilities