unit 2: counting methods - laval high · pdf file... how many ways can they be arranged if...
TRANSCRIPT
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You will be able to develop your critical
thinking skills related to counting by:
• Solving problems that involve theFundamental Counting Principle
• Understanding and simplifyingexpressions involving factorial notation
• Solving problems that involvepermutations
• Solving problems that involvecombinations
Unit 2: Counting Methods
2.1 COUNTING PRINCIPLES
Goal: Determine the Fundamental Counting Principle and use it to solve problems.
Example 1: Hannah plays on her school soccer team. The soccer uniform has three different sweaters: red,white, and black, and three different shorts: red, white, and black. How many different variations of the soccer uniform can the coach choose from for each game? Make a tree diagram.
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Try:
A toy manufacturer makes a wooden toy in three parts. Determine how many different
coloured toys can be produced? Make a tree diagram.
Part 1: The top part may be coloured red or blue
Part 2: The middle part may be orange, white, or black
Part 3: The bottom part may be yellow or green
Fundamental Counting Principle
If there are a ways to perform one task and b ways to perform another, then there are a × b ways
of performing both.
Consider a task made up of several stages. The fundamental counting principle states that if the
number of choices for the first stage is a , the number of choices for the second stage is b , the
number of choices for the third stage is c, etc.. then the number of ways in which
the task can be completed is a × b × c ×....
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Recap Example 1: Hannah plays on her school soccer team. The soccer uniform has:
• Three different sweaters: red, white, and black, and
• Three different shorts: red, white, and black.
How many different variations of the soccer uniform can the coach choose from for each game?
Use the Fundamental Counting Principle:
Number of uniform variations = _______ x _______ = ________
There are ___________ different variations of the soccer uniform to choose from.
Solve a Counting Problem by Extending the Fundamental Counting Principle
Example 2: A luggage lock opens with the correct threedigit code. Each wheel
rotates through the digits 0 to 9.
a. How many different threedigit codes are possible?
b. Suppose each digit can be used only once in a code. How many different codes are possible
when repetition is not allowed?
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Try:
A vehicle license plate consists of 3 letters followed by 3 digits. How many license plates are possible if:
a. there are no restrictions on the letters or digits used?
b. no letter may be repeated?
c. the first digit cannot be zero and no digits can be repeated?
Example 3: Solving a counting problem when the Fundamental Counting
Principle does not apply
A standard deck of cards contains 52 cards as shown.
Count the number of possibilities of drawing a single card and getting:
a. either a red face card or a king
b. either a club or a two
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Example 5: A bike lock opens with the correct fourdigit code set by rotating five wheels
through the digits 0 to 9.
a) How many different fourdigit codes are possible?
Number of different codes = ______ x ______ x ______ x ______ = ________
There are _______ different fourdigit codes.
b) Suppose each digit can be used only once in a code. How many different codes are possible
when repetition is not allowed?
Number of different codes = ______ x ______ x ______ x ______ = _______
There are _______ different fourdigit codes when the digits cannot repeat.
The Fundamental Counting Principle applies when tasks are related by the word 'AND'.
If tasks are related by the work 'OR':
• If the tasks are mutually exclusive, they involve two disjoint sets A and B:
• If the tasks are not mutually exclusive, they involve two sets that are not disjoint, C and D:
The Principle of Inclusion and Exclusion must be used to avoid counting elements in the
intersection of the two sets more than once.
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Example 7: A standard deck of cards contains 52 cards. Count the number of possibilities of drawing a single card and getting:
a) either a black face card or an ace
There are _________ ways to draw a single card and get either a black face card or an ace.
b) either a red card or a 10
There are _________ ways to draw a single card and get either a red card or a 10.
Practice:p. 73‐75 #2,3,6,7,8 ("or"), 9,11,14,16
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Example 6 Permutations with ConditionsJean, Kyle, Colin and Lori are to be arranged in a line from left to right.
a) How many ways can they be arranged?
b) How many ways can they be arranged if Kyle and Colin must be side by side?
c)How many ways can they be arranged if Jean must be at one end of the line?
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Example 7
To open the garage door of Mary's house, she uses a keypad
containing the digits 0 through 9. The password must be at least a
4 digit code to a maximum of 6 digits, and each digit can only be
used once in the code. How many different codes are possible?
Practice:
Pages 93‐95 1,3,5,8,9,10,13,14,15
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4. Generalize the pattern from the investigation to determine the number of permutations of:
a﴿ A, B, C, D, D b﴿ A, B, D, D, D c﴿ A, D, D, D, D
d﴿ A, B, B, C, C e﴿ A, A, A, B, B
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Example 1: Determine the number of permutations of all the letters in the following the words.
a﴿ STATISTICIAN b﴿ CANADA
Example 2: How many ways can the letters of the word CANADA be arranged if the first
letter must be N and the last letter must be C?
Example 3
Julie’s home is three blocks north and five blocks west of her school. How many routes can Julie take from home to school if she always travels either south or east?
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Section 2.52.6 Combinations
p. 109
Goal: Solve problems involving combinations
Example 1
a) If 5 sprinters compete in a race final, how many different ways can the medals for first,
second, and third place be awarded?
b) If 5 sprinters compete in a qualifying heat of a race, how many different ways can the sprinters qualify?
A permutation is an arrangement of elements in which the order of the arrangement is taken into
account. A combination is a selection of element in which the order of selection is NOT taken
into account.
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Example 2
There are 16 students in a class. Determine the number of ways in which four students can be
chosen to complete a survey.
Example 3: Solving a Combination Problem Using the Fundamental Counting Principle
a) The Athletic Council decides to form a subcommittee of 7 council members to look at how funds raised should be spent on sport activities in the school. There are a total of 15 athletic council members, 9 males and 6 females. The subcommittee must consist of exactly 3 females. Determine the number of ways of selecting the subcommittee.
b) A basketball coach has 5 guards and 7 forwards on his basketball team. In how many different ways can he select a starting team of two guards and three forwards?
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Example 4: Solving a Combination Problem by Considering Cases
a) A planning committee is to be formed for a schoolwide Earth Day program. There are 13 volunteers: 8 teachers and 5 students. How many ways can the principal choose a 4person committee that has at least 1 teacher?
b﴿ An allnight showing at a movie theatre is to consist of five movies. There are fourteen
different movies available, ten disaster movies and four horror movies. How many possible
schedules include at least four disaster movies?
Practice recognizing when the order matters or not
Say whether each of these involve a permutation ﴾P﴿ or a combination ﴾C﴿
a﴿ the number of ways that 3 horses out of 8 could end up first, second and third in a race
b﴿ the number of different hands of 5 face cards you could draw from a deck of 52 cards
c﴿ the number of ways a team could win 2 or 3 games out of the next 5
d﴿ the number of different messages you could send with three coloured flags out of a set of 10
flags
e﴿ the number of ways you can rearrange the letters in the word BASEBALL
f﴿ the number of ways to pick a group of 2 boys and 3 girls from a class of 14 boys and 15 girls
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Practice:P. 118‐120 #4, 5, 8‐11, 13, 15a, 19
Section 2.7 Solving Counting Problems
Example #1:
Mr. Counsel and some of his favourite students (it could be you!) are having a group photograph taken. There are three boys and five girls. The photographer wants the boys to sit together and the girls to sit together for one of the poses. How many ways can the students and teacher sit in a row of nine chairs for this pose?
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Example 2
At a car lot, six different coloured cars are to be parked close to the street. The cars are blue, red,
white, black, silver, and green.
a) How many ways can the cars be parked?
b) How many ways can the cars be parked such that the white and black cars are always next to each
other.
Example 3
Twelve students are running for student council. How many ways can a student council of three be
chosen?
Example 4
Twelve students are running for president, treasurer and secretary of the class. How many ways can
the three positions be filled?
Example 5
There are 14 women and 8 men who audition for an improv team.
a) How many different ways could the team be chosen if there are 3 women and 2 men chosen?
b) How many different ways could 5 people be chosen if at least 3 men are chosen.