04/19/23 MATH 106, Section 2 1

Section 2

Tree Diagrams

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04/19/23 MATH 106, Section 2 2

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.

• Let’s use a tree diagram (a pictorial representation) to get all of the possibilities.

HHH HHT HTH THH HTT THT TTH TTT

04/19/23 MATH 106, Section 2 3

What we mean by a “tree” …

• 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!

heads

tails

1st toss

heads

tails

heads

tails

2nd toss

heads

tails

heads

tailsheads

tails

heads

tails

3rd toss

Let’s do more examples …

04/19/23 MATH 106, Section 2 4

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?

red

khaki

shirts pants

white

blue

khaki

khaki

khaki

khaki

blue jeans

blue jeans

blue jeans

blue jeans

black

black

black

black

The number of different outfits is 12.

The answer is (of course) the same if we pick pants first to build the tree diagram.

#1

04/19/23 MATH 106, Section 2 5

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.

04/19/23 MATH 106, Section 2 6

A hot dog cart has the following condiments: ketchup, mustard, and relish. How many different “kinds” of hot dogs can you make?

yes

no

ketchup mustard relish

yes

no

yes

no

yes

noyes

no

yes

noyes

no

The number of different “kinds” of hot dogs can you make is 8.

#2

04/19/23 MATH 106, Section 2 7

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.

04/19/23 MATH 106, Section 2 8

Homework Hints:

In Section 2 Homework Problem #8,

In Section 2 Homework Problem #9,

In Section 2 Homework Problems #11 and #12,

don’t be confused or fooled by the way the sentence is worded. Consider “cream” possibilities and “sugar” possibilities separately.

consider each topping separately.

consider possibilities for die#1 followed by possibilities for die#2 in a tree diagram.