COUNTING METHODS AND PROBABILITY

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MGF1106
Unit Four Ex. 6.1 Counting Methods And
Probability Objective 1
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Exercise 6.1
Objective 1
To use the fundamental counting principle to
determine the number of ways a specified event
will occur
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The Fundamental Counting Principle
Given event E can occur in m ways, event F can
occur in n ways, event G can occur in p ways, and
so on. The number of ways that event E can
occur, followed by event F followed by event G
is determined in the following way.
Write three blanks to represent the three events.
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Place in each blank the number of ways the event
can occur.
m
n
p
Find the product of those numbers.
The answer is
mnp
The Fundamental Counting Principle can be
extended to any number of events.
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Find how many two-digit positive integers can be
written using the digits from 1,2,3,4,5 if
1) repetition of the digits is permitted.
Repetition means the same digit could be used
more than one time.
Draw two blanks since we want the number of two-
digit integers.
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In the first blank put the number of digits that
could be used.
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Find how many two-digit positive integers can be
written using the digits from 1,2,3,4,5 if
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Find how many two-digit positive integers can be
written using the digits from 1,2,3,4,5 if
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In the second blank put the number of digits that
could be used.
Now multiply the two numbers together.
The number of two-digit positive integers is 25.
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Find how many two-digit positive integers can be
written using the digits from 1,2,3,4,5 if
2) repetition of the digits is not permitted.
4
5
The number of two-digit positive integers is 20.
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Find how many two-digit positive integers can be
written using the digits from 1,2,3,4,5 if
3) repetition is permitted and the integer is
even.
Draw two blanks.
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Notice we want the number to be even. That means
the second digit has to be 2 or 4.
How many choices do we have for the second digit?
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Find how many two-digit positive integers can be
written using the digits from 1,2,3,4,5 if
3) repetition is permitted and the integer is
even.
2
5
10
Now write in the first blank the number of
choices for that digit.
Find the product of the two numbers.
The number of two-digit positive integers is 10.
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Find how many two-digit positive integers can be
written using the digits from 1,2,3,4,5 if
4) repetition is not permitted and the integer
is even.
2
4
8
Find the product.
The number of two-digit positive integers is 8.
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A student selects clothes that may be worn
together if desired. The clothing consists of 4
skirts, 5 blouses, and 2 sweaters. How many
different outfits are possible, if an outfit
consists of a skirt, blouse, and a sweater?
4
2
5
40
Need three blanks
Find the product.
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A club has 3 candidates for president, 5 for vice
president, and 4 for secretary-treasurer. How
many different executive councils can be elected?
3
5
4
Number of blanks
Multiply (3)(5)(4) 60
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How many three-digit positive integers can be
written using the digits 1, 2, 3, 4, 5, 6, 7, 8
and 9 with no digits repeated?
9
8
7
504
What is the number of choices for the first blank?
What is the number of choices for the second
blank?
What is the number of choices for the third
blank?
Find the product.
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How many seven-digit telephone numbers can be
formed if the first digit must be a 7?
1
10
10
10
10
10
10
106
How many digits can be placed in the first blank?
How many digits can be placed in the second blank?
How many digits can be placed in each of the
other blanks?
Form the product.
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How many four-digit odd numbers can be formed
using the digits in 1,2,3,4,5 if digits may be
repeated?
5
5
3
5
375
Since the number has to be odd, the number of
digits in the last blank should be decided first.
Use 1, 3, or 5
How many digits may be used in the other blanks?
Form the product.
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How many ways can Betty, Sue, John, Tim, Harold
arrange themselves in a row?
120
5
4
3
2
1
How many people can be in the first position?
How many people can be in the second position?
Third position?
Fourth position?
Last position?
Find the product.