Counting Principle
Jolie Martis is going to buy a new automobile. She has already chosen the make and model
of the car but she still has three more decisions to make.
1. Does she want standard or automatic transmission?
2. Does she want to have a cassette player or does she want a
compact disc player?
3. Does she want a silver, red or white exterior?
These three decisions are called independent events since one decision does not affect the
others. The tree diagram shown below illustrates all the different choices Jolie has in
making her final three decisions.
silver
cassette player
red
white
standard
silver
CD player
red
white
silver
cassette player
red
white
automatic
silver
CD player
red
white
One of Jolie’s choices is a red car with standard transmission and cassette player. There
are 11 other choices making a total of 12.
You can find the total number of choices that Jolie has without drawing a tree diagram.
Choices:
standard/automatic
cassette/CD player
silver/white/red
# of choices:
2
2
3
The total number of choices can be found be multiplying the number of choices for each
decision. This, the total number of choices is 2 • 2 • 3 or 12. This is an example of the
BASIC COUNTING PRINCIPLE.
Example 1: How many different 3-letter patterns can be formed using the letters x, y,
and z, if a letter can be used more than once?
Since each choice of letter is not affected by the previous choice, these are
independent events.
Letter:
# of choices:
1st
3
2nd
3
3rd
3
There are 3 • 3 • 3 or 27 possible patterns.
NOTE: Some applications involve dependent events. That is, the number of
choices for one even does affect other events.
Example 2: How many different 3-letter patterns can be formed using the letters x, y,
and z, if each letter is used exactly once?
After the first letter is chosen, it cannot be chosen again. So there are only
two choices for the second letter. Likewise, after the second choice is made,
there is only one choice for the third letter. These are dependent events.
Letter:
# of choices:
1st
3
2nd
2
3rd
1
There are 3 • 2 • 1 or 6 possible patterns.
NOTE: 3 • 2 • 1 can be written as 3! (! Is read as “factorial”)
Examples:
5! = 5 • 4 • 3 • 2 • 1 = 120
7! = 7 • 6 • 5 • 4 • 3 • 2 • 1 = 5 040
Example 3: How many 7-digit phone numbers can begin with the prefix 890?
Since each digit can be used any number of times, there are 10 choices for
each of the last four digits of the phone number. These are independent
events.
Digit in phone number:
# of choices:
4th
10
5th
10
6th
10
There are 10 • 10 • 10 • 10 = 10 4 or 10 000 possible phone numbers.
7th
10
Example 4: Suppose you live in a plane represent towns that are connected
by roads. Starting at any one town, how many different
routes are there so that you visit each town exactly once?
Let’s name the points A, B, C, D, and E. Since each town you
visit limits the number of towns left to visit, these are
dependent events.
Points:
# of choices:
1st
5
2nd
4
3rd
3
4th
2
5th
1
There are 5 • 4 • 3 • 2 • 1 = 5! or 120 possible routes you can take.
EXERCISES: (check if situations are independent or dependent events first, and then
solve accordingly)
1.
Describe the difference between independent and dependent events.
2.
Give two examples of independent events.
3.
Give two examples of dependent events.
4.
Are the results of tossing a coin several times independent events or dependent
events.
5.
Tell whether the events are independent or dependent.
a. Selecting a mystery book and a history book at the library.
b. Drawing cards from a deck to form a 5-card hand.
c. Selecting the color and model of a new automobile.
d. Choosing the color and size of a pair of pants.
e. Choosing a president, secretary and treasurer from the Social Club.
f. Choosing five numbers in a bingo game.
g. Choosing the winner and loser of a chess game.
h. Each of five people guess the total number of runs in a baseball game. They
write down the guess without telling what it is.
i. The numerals 0 through 9 are written on pieces of paper and placed in a jar.
Three of them are selected one after the other without replacing any pieces
of paper.
6.
How many different batting orders does a baseball team of nine players have if
the pitcher bats last?
7.
At FLECs, Darrin is taking six different classes. Assuming that each of these
classes is offered each period, how many different schedules might he have?
8.
The letters g, h, j, k and l are to be used to form 5-letter passwords for an office
security system. How many passwords can be formed if the letters can be used
more than once in any password?
9.
A store has 15 sofas, 12 lamps, and 10 tables at half price. How many different
combinations of a sofa, a lamp, and a table can be sold at half price?
10.
Draw a tree diagram to illustrate all the possibilities.
a. The possibilities for boys and girls in a family with two children.
b. The possibilities for boys and girls in a family with three children.
11.
A license plate must have two letters (not I or O) followed by three digits. The
last digit cannot be zero. How many possible plates are there?
12.
There are five roads form Amherst to Bedford, six from Bedford to Canso, and
three from Canso to Dartmouth. How many different routes are there from
Amherst to Dartmouth via Bedford and Canso?
13.
For a particular model of car, a car dealer offers 6 versions of that model, 18
body colors, and 7 upholstery colors. How many different possibilities are
available for that model?
14.
How many ways can six different books be arranged on a shelf?
15.
Three different colored dice are tossed. How many distinct outcomes can occur?
16.
How many ways can six books be arranged on a shelf if one of the books is a
dictionary and it must be on the end?
17.
Using the letters from the word equation, how many 5-letter patterns can be
formed in which q is followed immediately by u?
Answers:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
IE Æ no affect on another
DEÆ has affect on another
rolling dice, picking card and then replacing it
lottery, selecting members of a team from a group
IE
a) I
d) I
b) D
e) D
c) I or D
f) D
8! = 40 320 orders
6! = 720 schedules
5 x 5 x 5 x 5 x 5 = 3125
15 x 12 x 10 = 1800
a)
b)
B
B
11.
12.
13.
14.
15.
16.
G
G
B
B
G
B
g) D
h) I
i) D
G
G
B
G B
24 x 24 x 10 x 10 x 9 = 518 400
5 x 6 x 3 = 90
6 x 18 x 7 = 756
6 x 5 x 4 x 3 x 2 x 1 = 6! = 720
6 x 6 x 6 = 216
(5 x 4 x 3 x 2 x 1 ) + (1 x 5 x 4 x 3 x 2 x 1 ) = 240
B
G
G B G B
?
?
?
?
?
5 books to chose from for
this position
4 books to
chose from for
this position
3 books to
chose from for
this position
2 books to
chose from for
this position
1 books to
chose from for
this position
Dictionary
1 books to chose from for
this position (must be
dictionary)
?
5 books to
chose from for
this position
?
4 books to
chose from for
this position
?
3 books to
chose from for
this position
?
2 books to
chose from for
this position
17.
G
Dictionary
1 books to chose
from for this
position (must be
dictionary)
?
1 books to chose
from for this
position
(1 x 1 x 6 x 5 x 4) + (6 x 1 x 1 x 5 x 4) + (6 x 5 x 1 x 1 x 4) + (6 x 5 x 4 x 1 x 1) = 480
Q
1 letter to choose from
(must be Q)
U
1 letter to choose from
(must be U)
?
6 letters to choose
from
?
5 letters to choose
from
?
4 letters to choose
from
?
6 letters to choose
from
Q
1 letter to choose from
(must be Q)
U
1 letter to choose from
(must be U)
?
5 letters to choose
from
?
4 letters to choose
from
?
6 letters to choose
from
?
5 letters to choose
from
Q
1 letter to choose from
(must be Q)
U
1 letter to choose from
(must be U)
?
4 letters to choose
from
?
6 letters to choose
from
?
5 letters to choose
from
?
4 letters to choose
from
Q
1 letter to choose from
(must be Q)
U
1 letter to choose from
(must be U)