Section 2
Tree Diagrams
Questions about homework?
Submit homework!
3/24/2016
MATH 106, Section 2
1
Using “brute force” to solve problems
• Some simple problems can be solved using “brute force”
• List all possible arrangements and count them
• The hard part – making sure that the list includes each one
exactly once
• Don’t miss any
• No duplicates
• You are going to flip a coin three times. How many possible
outcomes are there? Let’s list them.
HHH
HHT
HTH
THH
HTT
THT
TTH
TTT
• Let’s use a tree diagram (a pictorial representation) to get all
of the possibilities.
3/24/2016
MATH 106, Section 2
2
What we mean by a “tree” …
1st toss
2nd toss
3rd toss
heads
heads
• A common point
called the root
• The lines are the
branches
• The more parts
there are, the taller
the tree is
• We will grow our
trees to the right!
tails
heads
heads
tails
tails
heads
heads
tails
tails
heads
tails
Let’s do more examples …
3/24/2016
MATH 106, Section 2
tails
3
#1
You have 4 shirts (red, white, blue, and khaki) and 3 pairs of pants
(khaki, black, and blue jeans). Construct a tree diagram to find how
many different outfits can you make. Does it matter whether you
pick shirts first or pants first to build the tree diagram?
shirts
pants
red
white
blue
khaki
3/24/2016
khaki
black
blue jeans
khaki
black
blue jeans
khaki
black
blue jeans
khaki
black
blue jeans
The number of different outfits is 12.
The answer is (of course) the same if
we pick pants first to build the tree
diagram.
MATH 106, Section 2
4
How many different outfits can you make, if you never have the same
color shirt and pants?
The number of different outfits without the same color shirt and
pants is 12 – 2 = 10.
3/24/2016
MATH 106, Section 2
5
#2
A hot dog cart has the following condiments: ketchup, mustard, and
relish. How many different “kinds” of hot dogs can you make?
ketchup
mustard
relish
yes
yes
no
yes
yes
no
no
The number of different “kinds”
of hot dogs can you make is 8.
yes
yes
no
yes
no
no
no
3/24/2016
MATH 106, Section 2
6
How many different “kinds” of hot dogs can you make with exactly 2
toppings?
The number of different “kinds” of hot dogs can you make with
exactly 2 toppings is 3.
3/24/2016
MATH 106, Section 2
7
Homework Hints:
In Section 2 Homework Problem #8,
don’t be confused or fooled by the way the sentence is
worded. Consider “cream” possibilities and “sugar”
possibilities separately.
In Section 2 Homework Problem #9,
consider each topping separately.
In Section 2 Homework Problems #11 and #12,
consider possibilities for die#1 followed by possibilities for
die#2 in a tree diagram.
3/24/2016
MATH 106, Section 2
8