mat 2720 discrete mathematics section 6.1 basic counting principles
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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
•Addition Principle
•Inclusion-Exclusion Principle
Analysis
License Plate
# of possible plates = ?Procedure: Step 1: Step 4:Step 2: Step 5:Step 3: Step 6:
LLL-DDD
Multiplication Principle
Suppose a procedure can be constructed by a series of steps
1Step
1 n ways
Step k
kn ways
3Step
3 n ways
2Step
2 n ways
Number of possible ways to complete the procedure is
1 2 kn n n
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.
# of possible ways = ?
EE Department
37 Professors
83 Students
Analysis
Pick a person to joint a university committee.
# of possible ways = ?
The 2 sets: :
EE Department
37 Professors
83 Students
Addition Principle
Number of possible element that can be selected from X1 or X2 or …or Xk is
OR
1X
1 n elements
2X
2 n elements
3X
3 n elements
kX
kn elements
1 2 kn n n
1 2 1 2k kX X X n n n
Example 4
A 6-person committee composed of A, B, C , D, E, and F is to select a chairperson, secretary, and treasurer.
Committee
chairperson
secretary
treasurer
A,B,C,D,E,F
Example 4 (a)
In how many ways can this be done?
Committee
chairperson
secretary
treasurer
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?
Committee
chairperson
secretary
treasurer
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?
Committee
chairperson
secretary
treasurer
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?
Committee
chairperson
secretary
treasurer
A,B,C,D,E,F
Recall: Intersection of Sets (1.1)The intersection of X and Y is defined as the set
| and X Y x x X x Y
X Y
X Y
X Y52
413
Recall: Intersection of Sets (1.1)The intersection of X and Y is defined as the set
| and X Y x x X x Y
X Y
1,2,3 , 3, 4,5
3
X Y
X Y
X Y52
413
Example 5
What is the relationship between
, , , and ?X Y X Y X Y
X Y
1,2,3
3, 4,5
3
1, 2,3,4,5
X X
Y 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?.
Committee
chairperson
secretary
treasurer
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 2720Discrete Mathematics
Section 6.2
Permutations and Combinations Part I
http://myhome.spu.edu/lauw
Example 1
6 persons are competing for 4 prizes. How many different outcomes are possible?
1st prize
CD EF
2nd prize 3rd prize 4th prize
AB
Step 1:
Step 2:
Step 3:
Step 4:
r-permutations
A r-permutation of n distinct objects
is an ordering of an r-element subset of
1 2, , , nx x x
1 2, , , nx x x
1st 2nd 3rd r - th
3x1x nx
2x
r-permutations
A r-permutation of n distinct objects
is an ordering of an r-element subset of
The number of all possible ordering:
1 2, , , nx x x
1 2, , , nx x x
1st
3x1x nx
2x
2nd 3rd r - th
( , )P n r
Example 1
6 persons are competing for 4 prizes. How many different outcomes are possible?
1st prize
CD EF
2nd prize 3rd prize 4th prize
AB
(6,4)P
Example 2
100 persons enter into a contest. How many possible ways to select the 1st, 2nd, and 3rd prize winner?
Example 3(b)
How many permutations of the letters A, B, C , D, E, and F are possible.
Note that, “permutations” means “6-permutations”.
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