Chapter 11
Objectives
11.1 Counting by systematic Listing
11.2 Counting Using Fundamental Counting Principle
11.3 Counting Using Permutations and Combinations.
11.1 Counting by Systematic Listing
The methods of counting presented in this section involve listing the possible results for a given task. This approach is practical only for fairly short lists.
When listing possibleresults, it is extremely important to use a systematic approach, or we are likely tomiss some results.
-Counting One-Part Tasks.
-Counting Using Product Tables for Two-Part Tasks
-Counting Using Tree Diagram for Multiple-Part Tasks
One-Part Tasks: The results of a simple one part tasks can often be listed easily.
Example
If the task is rolling a single fair die once. How many different outcomes are possible?
Let’s Do It!
Consider a club N with five members: N= {Alan, Bill, Cathy, David, Evelyn}. In how many ways can this group select a president (assuming all members are eligible)?
Product Tables for Two-Part Tasks
Example
Determine the number of different possible results when two ordinary dice are rolled.
Solution
To easily distinguish between the dice we assume one is red and the other is green. Then the task consists of two parts: 1) Roll a red die 2) Roll a green die
The product table below shows that there are 36 possible outcomes.
Let’s Do it!
a)Find the number of ways that club N= {Alan, Bill, Cathy, David, Evelyn}.can elect both a president and asecretary. Assume that all members are eligible, but that no one can hold both offices.
b)Find the number of ways that club Ncan select a president and a treasurer if the president must be afemale
c)Find the number of ways that club N can appoint a committee of two members to represent them at an association conference.
Tree Diagrams for Multiple-Part Tasks
A task that has more than two parts is not easy to analyze with a product table. Another helpful device is the tree diagram
Example
Find the number of three-digit numbers that can be written using digits from the set {1,2,3} assuming that
(a) Repeated digits are allowed and
(b) Repeated digits are not allowed.
Solution
The task of constructing such a number has three parts:
1. Select the first digit. 2. Select the second digit. 3. Select the third digit.
a)
b.
Example
Michelle Clayton’s computer printer allows for optional settings with a panel of four on-off switches in a row. How many different settings can she select if no two adjacentswitches can both be off?
Solution
We denote“on” and “off” with 1 and 0, respectively (a common practice). Notice that each time on the tree diagram that a switch is indicated as off 0, the next switch can only be on 1. This is to satisfy the restrictionthat no two adjacent switches can both be off.
Let’s Do it!
Refer to club N={Alan, Bill, Cathy, David, Evelyn}.Assuming all members are eligible, but that no one canhold more than one office, list and count the differentways the club could elect each group of officers if
a)A president, a secretary, and a treasurer, if the president and treasurer must be women
b)A president, a secretary, and a treasurer, if the presidentmust be a man and the other two must be women
Let’s Do it!
Arne, Bobbette, Chuck, and Deirdre have tickets for four reserved seats in a row at a concert. In how many different ways can they seat themselves so that Arne and Bobbette will sit next to each other?
End of section 11.1. Start your online homework on MyMathLab.
11.2 Counting Using the Fundamental Counting Principle
Figure below which shows all possible non-repeating three-digit numbers with digits from the set {1, 2, 3}.
Let’s Do It!
Find the number of three-digit numbers follows the given descriptions:
a) Digits can be repeated.
b) No digits can be repeated.
c) No two adjacent digits are the same.
Let’s Do It!
In some states, auto license plates have contained three letters followed by three digits.
How many such licenses are possible?
Let’s Do It!
Recall the club N={ Alan, Bill, Cathy, David, Evelyn} In how many ways could they do each of the following?
A) Line up all five members for a photograph
B) Schedule one member to work in the office on each of five different days, assuming members may work more than one day.
C) Select a male and a female to decorate for a party
D) Select two members, one to open their next meeting and another to close it, given that Bill will not be present
The Factorial Expression
Let’s Do It!
Evaluate:
4! (6-3)!
(4!).(5!) 6!/3!
(6/3)!22!50!
Let’s Do It!
A) Erika Berg has seven essays to include in her English 1A folder. In how many different orders can she arrange them?
B) Lynn Damme is taking thirteen preschoolers to the park. How many ways can thechildren line up, in single file, to board the van?
Counting Distinguishable Arrangements
In counting arrangements of objects that contain look-alikes, the normal factorial formula must be modified to find the number of truly different arrangements.
Example
The number of distinguishable arrangements of the letters of the word DAD is not 3!= 6 but rather 3!/2! = 3 .
The listing in the margin shows how the six total arrangements consist of just three groups of two, where the two in a given group look alike.
In general, the distinguishable arrangements can be counted as follows.
Let’s Do It!
Determine the number of distinguishable arrangements of the letters in each word.
(a) HEEDLESS(b) NOMINEE
End of section 11.2. Start your online homework on MyMathLab.
11.3 Using Permutations and Combinations
Combinations
Recall Club N = {Alan, Bill, Cathy, David, Evelyn}. In how many ways we can select 3 members to serve on a committee?
Note that all possible size 3 committee of club N are:
There are ten subsets of size 3, so ten is the number of three-member committees possible.
Let’s Do it!
Describe each combination by words then evaluate the combination
5C3
12C5
35C20
Let’s Do it!
I) A common form of poker involves hands (sets) of five cards each, dealt from a standard deck consisting of 52 different cards (illustrated in the margin). How many different 5-card hands are possible?
II) How many of the possible 5-card hands from a standard 52-card deck would consist of the following cards?
(a) Four clubs and one non-club
(b) Two face cards and three non-face cards
(c) Two red cards, two clubs, and a spade
Permutations
How many arrangements are there of five things taken three at a time?
The answer, by the fundamental counting principle, is 5.4.3=60. The factors begin with 5 and proceed downward, just as in a factorial product, but do not go all the way to 1.
We now generalize this idea.In the context of counting problems, arrangements are often called permutations;the number of permutations of n distinct things taken r at a time is denoted nPr.
How many arrangements are there of five things taken three at a time?
Answer: Using
5P3=5!/(5-3)!
= 5!/2!
= 60 this means there are 60 different ways of arranging 3 object selected from a set of 5 objects.
Let’s Do it!
Describe each permutation by words then evaluate the permutation.
8P4
24P12
15P15
Let’s Do it!
A) How many ways can president and vice president be determined in a club with twelve members?
B) First, second, and third prizes are to be awarded to three different people. If there are ten eligible candidates, how many outcomes are possible?
Example
Suppose certain account numbers are to consist of two letters followed by four digits and then three more letters, where repetitions of letters or digits are not allowed withinany of the three groups, but the last group of letters may contain one or both of thoseused in the first group. How many such accounts are possible?
Solution
The task of designing such a number consists of three parts:
1. Determine the first set of two letters.
2. Determine the set of four digits.
3. Determine the final set of three letters.
Each part requires an arrangement without repetitions, which is a permutation.Multiply together the results of the three parts.
Is it Permutation or Combination!!
Both permutations and combinations produce the number of ways of selecting r items from n items where repetitions are not allowed. Permutations apply to arrangements (where order is important), while combinations apply to subsets (where order is not important).
Let’s Do it!
Decide whether each object is a permutation or a combination.
A telephone number
A Social Security number
A hand of cards in poker
A committee of politicians
The “combination” on a student gym locker combination locks
An automobile license plate number
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