In a recent survey the followinginformation was obtained:34 people read Time, 35 peopleRead 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

Time (34) Newsweek (35) 30 11 22 6 3 14 4

16

People (37)a. How many people were surveyed? 106b,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 P (73) 35 a) 18

b) 72

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.

A C 1 Miami G Orlando 7 Lake City T 9

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 numbers4) Write the number in the blank. _4 •_3 _ = 125) Use Fundamental Counting PrincipleA1, 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 E2) __4 x __4 = 16 N N3)__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 987There are 2 one-digit numbers, 4 two-digitnumbers, and 4 “3digit” numbers. Since 2+ 4 + 4 =10, this suggeststhat an “or” decision likeone 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?

__ 3 = 3 (2, 4, or 6)

___4 X __3 =12(2, 4, or 6)

____4 X___3 X _3 =36(2, 4, or 6)

___4 X___3 X___2 X__ 3 =72 (2, 4, or 6)

_ 4 X___3 X___2 X___1 X__ 3 =72 (2, 4, or 6)

3 + 12 + 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?

___2 = 2( 1, 3) __ 3 X___ 2 = 6 (1, 3) ____3 X__ 2 X ___ 2 =12 (1, 3) ___3 X___ 2 X___1 X __ 2 =12 (1, 3)

3 + 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 ABYXZ ABYZX ABZXY ABZYX

BAXYZ BAXZY

BAYXZ BAYZX

BAZXY BAZYX

Notice that

(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.

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_____=

N 1 N 2 N 3 N 4 L 1 L 2 L 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?

_4 X_3 X_ 2 X_ 1 X_3 X_2 X__1 = 144N 1 N 2 N 3 N 4 L 1 L 2 L 3