§ 3.4

§ 3.4

Counting Principles

Larson & Farber, Elementary Statistics: Picturing the World, 3e 2

If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m· n. This rule can be extended for any number of events occurring in a sequence.

Fundamental Counting Principle

Example:A meal consists of a main dish, a side dish, and a dessert. How many different meals can be selected if there are 4 main dishes, 2 side dishes and 5 desserts available?

# of main dishes

# of side dishes

# of desserts

4 52 =There are 40 meals available.

40



Example:Two coins are flipped. How many different outcomes are there? List the sample space.

1st Coin Tossed

Start

Heads Tails

Heads Tails

2nd Coin Tossed

There are 2 2 = 4 different outcomes: {HH, HT, TH, TT}.

Heads Tails

2 ways to flip the coin

2 ways to flip the coin



Example:The access code to a house's security system consists of 5 digits. Each digit can be 0 through 9. How many different codes are available if a.) each digit can be repeated?b.) each digit can only be used once and not repeated?a.) Because each digit can be repeated, there are

10 choices for each of the 5 digits.

10 · 10 · 10 · 10 · 10 = 100,000 codes

b.) Because each digit cannot be repeated, there are 10 choices for the first digit, 9 choices left for the second digit, 8 for the third, 7 for the fourth and 6 for the fifth. 10 · 9 · 8 · 7 · 6 = 30,240 codes


Permutations

Example:How many different surveys are required to cover all possible question arrangements if there are 7 questions in a survey?

A permutation is an ordered arrangement of objects. The number of different permutations of n distinct objects is n!.

“n factorial”

n! = n · (n – 1)· (n – 2)· (n – 3)· …· 3· 2· 1

7! = 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5040 surveys


HOMEWORK

Page 157 1-11 ALL


Permutation of n Objects Taken r at a Time

The number of permutations of n elements taken r at a time is

8 5Pn rP 8 7 6 5 4 3 2 1=3 2 1

6720 ways

n rP# in the group # taken

from the group

! .( )!nn r

Example:You are required to read 5 books from a list of 8. In how many different orders can you do so?

8!(8 5)!

8!3!


Distinguishable Permutations

The number of distinguishable permutations of n objects, where n1 are one type, n2 are another type, and so on is

1 2 3 1 2 3

, where !! ! ! !

k

nn n n n

kn n n n n.

Example:Jessie wants to plant 10 plants along the sidewalk in her front yard. She has 3 rose bushes, 4 daffodils, and 3 lilies. In how many distinguishable ways can the plants be arranged?

10!3!4!3!

4,200 different ways to arrange the plants

10 9 8 7 6 5 4!3!4!3!


Example:You are required to read 5 books from a list of 8. In how many different ways can you do so if the order doesn’t matter?

A combination is a selection of r objects from a group of n things when order does not matter. The number of combinations of r objects selected from a group of n objects is

Combination of n Objects Taken r at a Time

! .( )! !n

n r rnC r# in the

collection # taken from the collection

8 58!=

3!5!C 8 7 6 5!=

3!5!

combinations=56


Application of Counting Principles

Example:In a state lottery, you must correctly select 6 numbers (in any order) out of 44 to win the grand prize.

a.) How many ways can 6 numbers be chosen from the 44 numbers?

b.) If you purchase one lottery ticket, what is the probability of winning the top prize?

a.)

b.) There is only one winning ticket, therefore,

44 644!

6!38!C 7,059,052 combinations

1(win)7059052

P 0.00000014



Homework

Page 157-159 #12-3412 72 14 6416 720 18 2420 25P9=741,354,768,00022 13!=6,227,020,800 24 50,40026 4845 28 20,358,52030a 6,7600,000 b 5,760,000 c 0.5032a 1000 b 800 c ½ 34a 56 b 56 c 112d. 0.067


homework

Page 159-160 35-40

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§ 3.4