PPT

```MAT 2720
Discrete Mathematics
Section 6.1
Basic Counting Principles
http://myhome.spu.edu/lauw
General Goals




Develop counting techniques.
Set up a framework for solving counting
problems.
The key is not (just) the correct answers.
The key is to explain to your audiences
how to get to the correct answers
(communications).
Goals

Basics of Counting
• Multiplication Principle
• Inclusion-Exclusion Principle
Example 1
LLL-DDD
# of possible plates = ?
Analysis
LLL-DDD
# of possible plates = ?
Procedure:
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
Multiplication Principle
Suppose a procedure can be constructed
by a series of steps
Step 1
Step 2
Step 3
Step k
n1 ways
n2 ways
n3 ways
nk ways
Number of possible ways to complete the procedure is
n1  n2 
 nk
Example 2(a)
Form a string of length 4 from the letters
A, B, C , D, E without repetitions.
How many possible strings?
Example 2(b)
Form a string of length 4 from the letters
A, B, C , D, E without repetitions.
How many possible strings begin with B?
Example 3
Pick a person to joint a university committee.
37
Professors
83
Students
EE Department
# of possible ways = ?
Analysis
Pick a person to joint a university committee.
37
Professors
83
Students
EE Department
# of possible ways = ?
The 2 sets:
:
X1
X2
X3
Xk
n1 elements
n2 elements
n3 elements
nk elements

Number of possible element that can be selected
from X1 or X2 or …or Xk is n1  n2   nk

OR
X1  X 2 
 X k  n1  n2 
 nk
Example 4
A 6-person committee composed of A, B,
C , D, E, and F is to select a chairperson,
secretary, and treasurer.
chairperson
secretary
treasurer
Committee
A,B,C,D,E,F
Example 4 (a)
In how many ways can this be done?
chairperson
secretary
treasurer
Committee
A,B,C,D,E,F
Example 4 (b)
In how many ways can this be done if
either A or B must be chairperson?
chairperson
secretary
treasurer
Committee
A,B,C,D,E,F
Example 4 (c)
In how many ways can this be done if E
must hold one of the offices?
chairperson
secretary
treasurer
Committee
A,B,C,D,E,F
Example 4 (d)
In how many ways can this be done if both
A and D must hold office?
chairperson
secretary
treasurer
Committee
A,B,C,D,E,F
Recall:
Intersection of Sets (1.1)
The intersection of X and Y is defined as
the set
X Y  x | x  X and x Y 
X Y
X
Y
Recall:
Intersection of Sets (1.1)
The intersection of X and Y is defined as
the set
X Y  x | x  X and x Y 
X  1, 2,3 , Y  3, 4,5
X Y
X  Y  3
4
1
2
X
3
Y
5
Example 5
What is the relationship between
X , Y , X Y , and X Y ?
X Y
X Y
4
1
2
X
3
Y
5
X  1, 2,3
X 
Y  3, 4,5
Y 
X  Y  3
X Y 
X  Y  1, 2,3, 4,5 X  Y 
Inclusion-Exclusion Principle
X Y  X  Y  X Y
X Y
X Y
X
Y
Example 4(e)
How many selections are there in which
either A or D or both are officers?.
chairperson
secretary
treasurer
Committee
A,B,C,D,E,F
Remarks on Presentations

Some explanations in words are required. In
particular, when using the Multiplication
Principle, use the “steps” to explain your
calculations

A conceptual diagram may be helpful.
MAT 2720
Discrete Mathematics
Section 6.2
Permutations and
Combinations Part I
http://myhome.spu.edu/lauw
Goals

Permutations and Combinations
• Definitions
• Formulas
• Binomial Coefficients
Example 1
6 persons are competing for 4 prizes. How
many different outcomes are possible?
1st prize
3rd prize
2nd prize
D
Step 1:
Step 2:
Step 3:
Step 4:
F
4th prize
C
E
B
A
r-permutations
A r-permutation of n distinct objects x1, x2 ,
is an ordering of an r-element subset of
x1, x2 , , xn 
1st
2nd
x1
x2
3rd
x3
r - th
xn
, xn
r-permutations
A r-permutation of n distinct objects x1, x2 ,
is an ordering of an r-element subset of
x1, x2 , , xn 
The number of all possible ordering: P(n, r )
1st
2nd
x1
x2
3rd
x3
r - th
xn
, xn
Example 1
6 persons are competing for 4 prizes. How
many different outcomes are possible?
1st prize
D
P(6, 4) 
3rd prize
2nd prize
F
4th prize
C
E
B
A
Theorem
P(n, r )  n  (n  1)  (n  2)
n!

(n  r )!
1st
2nd
x1
x2
3rd
x3
(n  r  1)
r - th
xn
Example 2
100 persons enter into a contest. How
many possible ways to select the 1st, 2nd,
and 3rd prize winner?
Example 3(a)
How many 3-permutations of the letters A,
B, C , D, E, and F are possible?
Example 3(b)
How many permutations of the letters A, B,
C , D, E, and F are possible.
Note that, “permutations” means “6permutations”.
Example 3(c)
How many permutations of the letters A, B,
C , D, E, and F contains the substring
DEF?
Example 3(d)
How many permutations of the letters A, B,
C , D, E, and F contains the letters D, E,
and F together in any order?
```

21 cards

15 cards

14 cards

31 cards

12 cards