Download File - Mrs. Hille`s FunZone

Tree Diagram
A tree diagram is a segmented
graph in the shape of a tree in
which no branch leads from
any vertex back to itself. Each
path through it represents a
mutually exclusive event.
Calvary Christian Academy is
having an election of student
officers. Three students are
running for president—Juan,
Pam, and Jeff. There are two
candidates for vice president—
Doyle and Julianne. How many
different ways are there to fill
the offices?
Example 1
Find the number of ways that
a student can select a twodigit number if the first digit
must be odd and the second
digit must be less than five.
possible first digit—1, 3, 5, 7, 9
possible second digit—
0, 1, 2, 3, 4
1
3
5
01234
01234
7
9
01234
01234
01234
10, 11, 12, 13, 14
30, 31, 32, 33, 34
50, 51, 52, 53, 54
70, 71, 72, 73, 74
90, 91, 92, 93, 94
There are 25 different
two-digit numbers.
Example
Make a tree diagram to find
the number of combinations
of three pairs of pants, three
coats, and four shirts.
36
Example
Make a tree diagram to find
the number of possible
milkshakes that could be
ordered if chocolate and
vanilla shakes are available
in small, medium, and large.
6
Example
Make a tree diagram to find
the number of ways to make
fifty cents in change using
nickels, dimes, and
quarters.
10
Fundamental Principle of
Counting
If there are p ways that a first
choice can be made and q
ways that a second choice
can be made, then there are
p × q ways to make the first
choice followed by the
second choice.
Example 2
Reid has five dress shirts
and four ties. How many
different shirt-and-tie
combinations are possible?
5 × 4 = 20
Example 3
How many different twodigit counting numbers can
be formed if the first digit
must be a nonzero even
digit and the second digit
must be less than seven but
greater than zero?
There are four choices
(2, 4, 6, 8) for the first digit
and six choices
(1, 2, 3, 4, 5, 6) for the
second digit.
By the Fundamental
Principle of Counting there
are 4 x 6 = 24 such numbers.
Example 4
Mr. Dillard is buying a new car.
He has the options given in the
following table to choose from.
How many different options
does he have? If he chooses a
white exterior, how many
combinations does he have on
the remaining options?
AM/FM automatic
red
black
white
blue
+ CD
black
gray
+ DVD
silver
manual
How many different options
does he have?
4 × 3 × 3 × 2 = 72
If he chooses a white exterior,
how many combinations does
he have on the remaining
options?
1 × 3 × 3 × 2 = 18
Example
Use the Fundamental
Principle of Counting to find
the number of possible
three-digit area codes if the
first number cannot be
0 or 1.
800
Example
How many different license
plates are possible if three
letters must be followed by
three numbers?
17,576,000
Example
How many different license
plates are possible if none of
the letters or numbers can
repeat?
11,232,000
Example
How many ways can a family
of four line up for a
photograph?
24
Example
How many combinations are
possible on a school locker
if the lock consists of the
numbers 1 to 40 and the
combination is a three-digit
sequence of numbers?
64,000
Example
How many combinations are
possible if no two
consecutive numbers are the
same?
60,840
Example
How many ways can you
seat five couples in a row of
ten chairs, assuming, of
course, that each couple is
seated together?
3,840