Chapter 8Chapter 8Counting Principles: Counting Principles: Further Probability Further Probability

TopicsTopicsSection 8.1Section 8.1

The Multiplication The Multiplication Principle;Principle;

PermutationsPermutations

Warm – Up for Sections 8.1 Warm – Up for Sections 8.1 and 8.2and 8.2

A certain game at an amusement card A certain game at an amusement card consists of a person spinning a spinner, consists of a person spinning a spinner, choosing a card, and then tossing an choosing a card, and then tossing an unbiased coin. Prizes are awarded based unbiased coin. Prizes are awarded based on the combination created from on the combination created from performing each of the three tasks.performing each of the three tasks.The spinner has three equal areas The spinner has three equal areas represented by Purple, Gold, and Red; the represented by Purple, Gold, and Red; the cards to choose from include a King, cards to choose from include a King, Queen, and Joker; and the coin has a Queen, and Joker; and the coin has a Crown on one side and a Donkey on the Crown on one side and a Donkey on the other.other.How many possible outcomes are there?How many possible outcomes are there?If the order in which the tasks were If the order in which the tasks were performed made a difference, would there performed made a difference, would there be more outcomes or fewer outcomes?be more outcomes or fewer outcomes?

Alice can’t decide what to wear Alice can’t decide what to wear between a pair of shorts, a pair of between a pair of shorts, a pair of pants, and a skirt. She has four tops pants, and a skirt. She has four tops that will go with all three pieces: that will go with all three pieces: one red, one black, one white, and one red, one black, one white, and one striped.one striped.

How many different outfits could How many different outfits could Alice create from these items of Alice create from these items of clothing?clothing?

Bottom Top Outfit

Shorts

Pants

Skirt

RedBlackWhiteStriped

Red BlackWhiteStriped

RedBlackWhiteStriped

Shorts, Red Top

Shorts, Black TopShorts, White Top

Shorts, Striped Top

If the tree diagram is finished, how many outfits will she have?

12 outfits!!

Tree diagrams are not often Tree diagrams are not often convenient, or practical, to use when convenient, or practical, to use when determining the number of outcomes determining the number of outcomes that are possible.that are possible.

Rather than using a tree diagram to Rather than using a tree diagram to find the number of outfits that Alice find the number of outfits that Alice had to choose from, we could have had to choose from, we could have used a general principle of counting: used a general principle of counting: the multiplication principle. the multiplication principle.

Multiplication PrincipleMultiplication Principle

Alice can’t decide what to wear between a Alice can’t decide what to wear between a pair of shorts, a pair of pants, and a skirt. pair of shorts, a pair of pants, and a skirt. She has four tops that will go with all three She has four tops that will go with all three pieces: one red, one black, one white, and pieces: one red, one black, one white, and one striped.one striped.

How many different outfits could Alice How many different outfits could Alice create from these items of clothing?create from these items of clothing?

Using the multiplication principle, we multiply the Using the multiplication principle, we multiply the number of options she has for what to wear on number of options she has for what to wear on “bottom” and the number of options she has for what “bottom” and the number of options she has for what to wear on “top”.to wear on “top”.

3 bottoms • 4 tops = 12 outfits3 bottoms • 4 tops = 12 outfits

A product can be shipped by four A product can be shipped by four airlines and each airline can ship airlines and each airline can ship via three different routes. How via three different routes. How many distinct ways exist to ship the many distinct ways exist to ship the product?product?

How many different license plates can How many different license plates can be made if each license plate is to be made if each license plate is to consist of three letters followed by consist of three letters followed by three digits and replacement is three digits and replacement is allowed?allowed?

___ • ___ • ___ • ___ • ___ • ___ ___ • ___ • ___ • ___ • ___ • ___

L L L D D DL L L D D D

If replacement is not allowed?If replacement is not allowed?

26 26 26 10 10

10 = 26 ³ • 10 ³

= 17, 576, 000

___ • ___ • ___ • ___ • ___ • ___ • ___ • ___ • ___ • ___ • ___ ___ L L L D D L L L D D D D

26 25 24 10 9 8 = 11, 232, 000

How many different license plates How many different license plates can be made if each license plate can be made if each license plate begins with 63 followed by three begins with 63 followed by three letters and two digits?letters and two digits?

How many different social security How many different social security numbers are possible if the first digit numbers are possible if the first digit may not be zero?may not be zero?

Marie is planning her schedule for next Marie is planning her schedule for next semester. She must take the following five semester. She must take the following five courses: English, history, geology, courses: English, history, geology, psychology, and mathematics.psychology, and mathematics.

a.) In how many different ways can Marie a.) In how many different ways can Marie arrange arrange her schedule of courses?her schedule of courses?

b.) How many of these schedules have b.) How many of these schedules have mathematics mathematics listed first?listed first?

You are given the set of digits {1, 3, 4, 5, You are given the set of digits {1, 3, 4, 5, 6}.6}.

a.) How many three-digit numbers can a.) How many three-digit numbers can be formed?be formed?

b.) How many three-digits numbers can b.) How many three-digits numbers can be formed if be formed if the number must be even?the number must be even?

c.) How many three-digits numbers can c.) How many three-digits numbers can be formed if be formed if the number must be even and no the number must be even and no repetition of repetition of digits is allowed?digits is allowed?

A certain Math 110 teacher has A certain Math 110 teacher has individual photos of each of her three individual photos of each of her three dogs: Indy, Sam, and Jake. In how dogs: Indy, Sam, and Jake. In how many ways can she arrange these many ways can she arrange these photos in a row on her desk?photos in a row on her desk?

Factorial NotationFactorial Notation

If seven people board an airplane If seven people board an airplane and there are nine aisle seats, in how and there are nine aisle seats, in how many ways can the people be seated many ways can the people be seated if they all choose aisle seats?if they all choose aisle seats?

PermutationsPermutations

A A permutationpermutation of of rr (where (where rr ≥ 1) ≥ 1) elements from a set of elements from a set of nn elements is any elements is any specific ordering or arrangement, specific ordering or arrangement, without repetitionwithout repetition, of the , of the rr elements. elements.

Each rearrangement of the Each rearrangement of the rr elements is elements is a different permutation.a different permutation.

Permutations are denoted by Permutations are denoted by nPrnPr or or P(n, r)P(n, r)

Clue words: arrangement, schedule, Clue words: arrangement, schedule, order, awards, officersorder, awards, officers

A disc jockey can play eight records A disc jockey can play eight records in a 30-minute segment of her show. in a 30-minute segment of her show. For a particular 30-minute segment, For a particular 30-minute segment, she has 12 records to select from. In she has 12 records to select from. In how many ways can she arrange her how many ways can she arrange her program for the particular segment?program for the particular segment?

A chairperson and vice-chairperson A chairperson and vice-chairperson are to be selected from a group of are to be selected from a group of nine eligible people. In how many nine eligible people. In how many ways can this be done?ways can this be done?

Distinguishable Distinguishable PermutationsPermutations

If the If the nn objects in a permutation are objects in a permutation are not all distinguishable – that is, if there not all distinguishable – that is, if there so many of type 1, so many of type 2, so many of type 1, so many of type 2, and so on for and so on for rr different types, then different types, then the number of distinguishable the number of distinguishable permutations ispermutations is

n! .n! . n ! n ! ••• nn ! n ! ••• n ! !

1 r2

How many distinct arrangements can How many distinct arrangements can be formed from all the letters of be formed from all the letters of SHELTONSTATE?SHELTONSTATE?Step 1: Count the number of letters in the word, including repeats.

12 letters

Step 2: Count the number of repetitious letters and the number of times each letter repeats.

S : 2 repeats E : 2 repeats T : 3 repeats

Solution: 12! . 2! 2! 3!

= 19, 958, 000

In how many distinct ways can the In how many distinct ways can the letters of MATHEMATICS be letters of MATHEMATICS be arranged?arranged?

In how many distinct ways can the In how many distinct ways can the letters of BUCCANEERS be arranged?letters of BUCCANEERS be arranged?