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 number of decisions.

2) Draw a blank (____) for each.

3) Determine # of choices for each.

4) Write the number in the blank.

5) Use the Fundamental Counting Principle

1) 2 : Choosing a letter and a number

2) _____ _____

3) 4 letters, 3_numbers

4) 4__ 3__ 5) 4 x 3 = 12_ possible routesA1,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 =

west east

6

2) 4 x 4 =

north north

16

3) 9 x 9 =

enter leave

81

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

• 6 x 5 x 4 =

100’s 10’s 1’s

120

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 =

1000’s 100’s 10’s 1’s

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 =

1000’s 100’s 10’s 1’s

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

• 5 x 5 x 5 =

100’s 10’s 1’s

125

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 : , , , _

color fabric collors with/without

6 x 3 x 4 x 2 =

144

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

(Also 9! Called 9 Factorial)

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 2 =

1st 2nd 3rd 4th 5th

4 x ___ x ___ x ___ x 2 =

1st 2nd 3rd 4th 5th

4 x 3 x 2 x 1 x 2 =

1st 2nd 3rd 4th 5th

48

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 __ =

1st 2nd 3rd 4th 5th 6th

1 x 1 x 4 x __ x __ x __ =

1st 2nd 3rd 4th 5th 6th

1 x 1 x 4 x 3 x 2 x 1 =

1st 2nd 3rd 4th 5th 6th

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

2 x x __ x __ x __ x __ =

1st 2nd 3rd 4th 5th 6th

2 x 5 x x __ x __ x __ =

1st 2nd 3rd 4th 5th 6th

2 x 5 x 4 x 3 x 2 x 1 =

1st 2nd 3rd 4th 5th 6th

240

NOTE: From the digits 7, 8, 9, you can form 10odd numbers containingone or more digits if nodigit may be repeated ina number.

Since the numbers areodd, there are twochoices for the unitsdigit, 7 or 9.In this case, the numbersmay contain one, two, orthree digits.

1digit 7 92digit 79 87 89 973digit 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 moredigits, 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 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 =

+ + + + =

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) X, Y, Z can be arranged in 3!, or 6 ways; and(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

4 X___ X___X____X 3 X ____X___

ALG I ALG 2 ALG 3 ALG 4 GEOM I GEOM 2 GEOM 3

4 X 3 X 2 X 1 X 3 X 2 X 1 __

ALG I ALG 2 ALG 3 ALG 4 GEOM I GEOM 2 GEOM 3

= 144

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

4 X___ X___X____X 3 X ____X___

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

4 X 3 X 2 X 1 X 3 X 2 X 1 _

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

= 144