In a recent survey the following
information was obtained:
34 people read Time, 35 people
Read Newsweek, 37 read People,
9Read Time and Newsweek, 7
read Newsweek and People, 17
read Time and People, 3 read all
of these and 30 read none.
Time
Newsweek
People
Time (34)
Newsweek (35)
30
3
People (37)
Time (34)
Newsweek (35)
30
6
3
14
4
People (37)
d. How many people read exactly 2 of the magazines?
24
30
Time (34)
Newsweek (35)
11
22
6
3
14
4
16
People (37)
a. How many people were surveyed?
106
b,c.How many people read ONLY Time? ONLY People?
11
16
EX:  Of 180 people surveyed, 90 drink Coke,
73 drink Pepsi, and 35 drink neither
a) How many drink
both?
b) How many drink
ONLY Coke?
180
(90) C
35
a) 18
b) 72
P (73)
16.1 Fundamental Counting Principle
OBJ:  To find the number of possible
arrangements of objects by using the
Fundamental Counting Principle
DEF:  Fundamental Counting Principle
If one choice can be made in a ways and a
second choice can be made in b ways, then
the choices in order can be made in a x b
different ways.
EX:  A truck driver must drive from Miami to
Orlando and then continue on to Lake City.
There are 4 different routes that he can take
from Miami to Orlando and 3 different routes
from Orlando to Lake City.
Miami
A
C
G
T
1
Orlando 7
9
Lake City
Strategy for Problem Solving:
1) Determine the # of decisions. _2 Choosing a letter
and number.
2) Draw a blank (____) for each. _____•_____
3) Determine # of choices . letters numbers
4) Write the number in the blank. _4 •_3 _ = 12
5) Use Fundamental Counting Principle
A1, A7, A9;
C1, C7, C9;
G1, G7, G9;
T1, T7, T9
EX:  A park has nine gates—three on the
west side, four on the north side, and two on
the east side.
In how many different ways
can you :
1) enter the park from the
west side and later leave
from the east side?
2) enter from the north and
later exit from the north?
3)enter the park and later
leave the park?
_2: Choosing an entrance
and an exit gate
1) __3 x __2 = 6
W
E
2) __4 x __4 = 16
N
N
3)__9 x __9 = 81
Enter Exit
EX:  How many three-digit numbers can be formed
from the 6 digits: 1, 2, 6, 7, 8, 9 if no digit may be
repeated in a number
• _3 : Choosing a 100’s, 10’s, and1’s digit
• ____ x _____ x _____
100’s 10’s
1’s
• __6 x __5 x __4 = 120
100’s 10’s
1’s
EX:  How many four-digit numbers can be formed
from the digits 1, 2, 4, 5, 7, 8, 9
if no digit may be
repeated in a number?
_4: Choosing a
1000’s,100’s,10’s,1’s
digit
_7 x _6 x _5 x _4 =
840
If a digit may be repeated
in a number?
_4: Choosing a
1000’s,100’s,10’s,1’s
digit
_7 x _7 x _7 x _7 =
2401
EX:  How many three-digit numbers can be
formed from the digits 2, 4, 6, 8, 9 if a digit may be
repeated in a number?
• _3 : Choosing a 100’s, 10’s, and1’s digit
• ___ x ___ x ___
100’s 10’s 1’s
• _5 x _5 x _5 =125
100’s 10’s 1’s
EX:  A manufacturer makes sweaters in 6 different colors.
Each sweater is available with choices of 3 fabrics, 4 kinds of
collars, and with or without buttons.
How many different sweaters does the
manufacturer make?
__: _____ x ______ x ______ x ______ =
,
,
,
_
EX:  A manufacturer makes sweaters in 6 different colors.
Each sweater is available with choices of 3 fabrics, 4 kinds of
collars, and with or without buttons.
How many different sweaters does the manufacturer
make?
_4: __ 6 x __3 x __4 x __ 2 = 144
colors fabrics collars buttons
EX:  Find the number of possible batting
orders for the nine starting players on a
baseball team?
_9 decisions
_9 x _8 x _7 x _6 x _5 x _4 x _3 x _2 x _1
362,880
16.2 Conditional Permutations
OBJ:  To find the number of
permutations of objects when conditions
are attached to the arrangement.
DEF:  Permutation
An arrangement of objects in a definite order
EX:  How many permutations of all the letters
in the word MONEY end with either the letter E
or the letter y?
Choose the 5th letter, either a E or Y
___ x ___ x ___ x ___ x ___ =
EX:  How many permutations of all the letters
in the word MONEY end with either the letter E
or the letter y?
Choose the 5th letter, either a E or Y
___ x ___x ___ x ___ x _2
_4 x _3 x _2 x _1 x _2 = 48
EX:  How many permutations of all the letters
in PATRON begin with NO?
Choose the 1st two letters as NO
__ x __ x __ x __ x __ x __ =
EX:  How many permutations of all the letters
in PATRON begin with NO?
Choose the 1st two letters as NO
1 x 1 x _x _x _x _
1 x 1 x 4 x 3 x 2 x 1 = 24
EX:  How many permutations of all the letters in
PATRON begin with either N or O?
Choose the 1st letter, either N or O
__ x __ x __ x __ x __ x __ =
EX:  How many permutations of all the letters in
PATRON begin with either N or O?
Choose the 1st letter, either N or O
2 x _ x __ x __ x __ x _
2 x 5 x 4 x 3 x 2 x 1 = 240
NOTE: From the digits 7,
8, 9, you can form 10
odd numbers
containing one or more
digits if no digit may be
repeated in a number.
Since the numbers are
odd, there are two
choices for the units
digit, 7 or 9. In this
case, the numbers may
contain one, two, or
three digits.
_2_
(7, 9)
1digit 7
9
_2_•_2_
(7,9)
2digit 79 87 89 97
_2_ • _1_• _2_
(7, 9)
3digit 789 879 897 987
There are 2 one-digit
numbers, 4 two-digit
numbers, and 4 “3
digit” numbers. Since 2
+ 4 + 4 =10, this suggests
that an “or” decision like
one or more digits,
involves addition.
EX:  How many even numbers containing one or more
digits can be formed from 2, 3, 4, 5, 6 if no digit may be
repeated in a number?
Note : there are three choices for a units digit: 2, 4, or 6.
_______ =
________X________=
_________X________X________=
________X________X________X_______=
_____X_______X________X________X________=
+
+
+
+
=
EX:  How many even numbers containing one or more
digits can be formed from 2, 3, 4, 5, 6 if no digit may be
repeated in a number?
___4
_4
3 +
X___3
12
+
__ 3
=3
(2, 4, or 6)
___4
X __3
=12
(2, 4, or 6)
____4
X___3
X _3
=36
(2, 4, or 6)
X___3
X___2
X__ 3
=72
(2, 4, or 6)
X___2
X___1
X__ 3
=72
(2, 4, or 6)
36
+
72
+
72
= 195
EX:  How many odd numbers containing one or more
digits can be formed from 1, 2, 3, 4 if no digit can be
repeated in a number?
_______ =
________X________ =
_________X________X________ =
________X________X________X_______ =
+
+
+
=
EX:  How many odd numbers containing one or more
digits can be formed from 1, 2, 3, 4 if no digit can be
repeated in a number?
__ 3
____3
X__ 2
X___ 2
X___1
___3
3
+
6
+
12
+
___2
( 1, 3)
X___ 2
(1, 3)
X ___ 2
(1, 3)
X __ 2
(1, 3)
12
=2
=6
=12
=12
= 33
NOTE: In some situations, the total number of permutations is the
product of two or more numbers of permutations. For example,
there are 12 permutations of A, B, X, Y, Z with A, B to the left “and”
X, Y, Z to the right.
ABXYZ
ABXZY
Notice that
ABYXZ
ABYZX
ABZXY
ABZYX
BAXYZ
BAXZY
BAYXZ
BAYZX
(1) A, B can be arranged in 2!,
or 2 ways; (2! = 2 x 1)
(2) X, Y, Z can be arranged in
3!, or 6 ways; 3! = 3 x 2 x 1)
(3) A, B, X, Y, Z can be
arranged in 2! x 3!, or 12
ways.
An “and” decision involves
multiplication.
BAZXY
BAZYX
EX:  Four different algebra books and three different geometry
books are to be displayed on a shelf with the algebra books together
and to the left of the geometry books. How many such
arrangements are possible?
___X____X___X____X____X____X_____=
ALG I ALG 2 ALG 3 ALG 4 GEOM I GEOM 2 GEOM 3
EX:  Four different algebra books and three different geometry
books are to be displayed on a shelf with the algebra books together
and to the left of the geometry books. How many such
arrangements are possible?
_ 4 X_3 X 2 X_1 X__3 X__2 X__1 =144
ALG 1 ALG 2 ALG 3 ALG 4 GEOM I GEOM 2 GEOM 3
EX:  How many permutations of 1, A, 2, B, 3, C,
4 have all the letters together and to the right of
the digits?
___X____X___X____X____X____X_____=
N1 N2 N3 N4 L1 L2
L3
EX:  How many permutations of 1, A, 2, B, 3, C,
4 have all the letters together and to the right of
the digits?
_4 X_3 X_ 2 X_ 1 X_3 X_2 X__1 = 144
N1 N2 N3 N4 L1 L2 L3