Counting Outcomes

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Lesson 14-1

• Counting Outcomes

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Transparency 3-1
5-Minute Check on Chapter 2

• Evaluate 42 - x - 7 if x -3
• Find 4.1 ? (-0.5)
• Simplify each expression
• 3. 8(-2c 5) 9c
  4. (36d 18) / (-9)
• A bag of lollipops has 10 red, 15 green, and 15
  yellow lollipops. If one is chosen at random,
  what is the probability that it is not green?
• Which of the
  following is a true statement

Standardized Test Practice
8/4 lt 4/8
-4/8 lt -8/4
-4/8 gt -8/4
-4/8 gt 4/8
A
C
B
D
Click the mouse button or press the Space Bar to
display the answers.
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Objectives

• Count outcomes using a tree diagram
• Count outcomes using the Fundamental Counting
  Principle

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Vocabulary

• Tree diagram
• Sample space
• Event
• Fundamental Counting Principle
• Factorial

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Tree Diagram

• To map out all possible combinations of things, a
  tree diagram is useful to visually see why the
  Fundamental Counting Principle works.

Chocolate Cake
RoastBeef
Apple Pie
Spinach Salad
Chocolate Cake
Salmon
Apple Pie
Chocolate Cake
RoastBeef
3?2?2 12
Apple Pie
The BigMeal
Shrimp Salad
Chocolate Cake
Salmon
Apple Pie
Chocolate Cake
RoastBeef
Apple Pie
House Salad
Chocolate Cake
Salmon
Apple Pie
12 Different Combinations of Salads, Meal, and
Desert
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Factorials

• n!, read n-factorial, is defined by the
  followingn?(n-1)?(n-2)? ?3?2?1the product
  of every number between n and 1
• Examples5! 5 ?4 ?3 ?2 ?1 120 7! 7
  ?6 ? 5 ?4 ?3 ?2 ?1 5040
• Remember too 5! 5 ?4!
  7! 7 ?6 ?5!
• (Useful in dividing factorials)

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Example 1
At football games, a student concession stand
sells sandwiches on either wheat or rye bread.
The sandwiches come with salami, turkey, or ham,
and either chips, a brownie, or fruit. Use a tree
diagram to determine the number of possible
sandwich combinations.
Answer The tree diagram shows that there are 18
possible combinations.
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Example 2
The Too Cheap computer company sells custom made
personal computers. Customers have a choice of 11
different hard drives, 6 different keyboards, 4
different mice, and 4 different monitors. How
many different custom computers can you order?
Multiply to find the number of custom computers.
Answer The number of different custom
computers is 1056.
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Example 3
There are 8 students in the Algebra Club at
Central High School. The students want to stand
in a line for their yearbook picture. How many
different ways could the 8 students stand for
their picture?
The number of ways to arrange the students can be
found by multiplying the number of choices for
each position.
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Example 3 cont

• There are eight people from which to choose for
  the first position.
• After choosing a person for the first position,
  there are seven people left from which to choose
  for the second position.

• There are now six choices for the third position.

• This process continues until there is only one
  choice left for the last position.

Let n represent the number of arrangements.
Answer There are 40,320 different ways they
could stand.
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Example 4
Find the value of 9!.
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Example 5a
Jill and Miranda are going to a National Park for
their vacation. Near the campground where they
are staying, there are 8 hiking trails. How
many different ways can they hike all of the
trails if they hike each trail only once?
Use a factorial.
Answer There are 40,320 ways in which Jill and
Miranda can hike all 8 trails.
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Example 5b
Jill and Miranda are going to a National Park for
their vacation. Near the campground where they
are staying, there are 8 hiking trails. If they
only have time to hike on 5 of the trails, how
many ways can they do this?
Use the Fundamental Counting Principle to find
the sample space.
Answer There are 6720 ways that Jill and
Miranda can hike 5 of the trails.
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Summary Homework

• Summary
• Use a tree diagram to make a list of possible
  outcomes
• If an event M can occur m ways and is followed by
  an event N that can occur n ways, the event M
  followed by event N can occur m?n ways
• Homework
• none