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Finite MathematicsMAT 141: Chapter 8 NotesDavid J. GischApril 7, 2012

The Multiplication Principle; Permutations

Multiplication Principle Multiplication Principle

• You can think of the multiplication principle as counting the branches (combinations) on a tree diagram.

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Multiplication PrincipleExample 8.1.1: You have 12 pairs of shoes, 5 pairs of pants and 6 different tops. How many outfits can you make?

Multiplication PrincipleExample 8.1.2: A housing builder provides the following options:• Deck or patio.• Hardwood, carpet or linoleum• Granite, quartz, or compound• Expensive landscaping, cheap landscaping or no

landscaping.How many different combinations do they offer?

Multiplication PrincipleExample 8.1.3: How many possible 10 digit phone numbers can there be if the first and fourth digit cannot be a zero?

Multiplication Principle (Dependence)

Example 8.1.4: For the Iowa Lottery one must choose 5 out of 59 numbers for the white balls; then choose 1 out of 35 numbers for the Powerball. How many possible tickets are there?

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Multiplication Principle (Dependence)

Example 8.1.5: You have 6 books and want to place them on a shelf. In how many ways can you organize the books?

Factorial!• As you saw in the last few examples we often end up

multiply successive terms until we reach one.• There is a mathematical notation for this.

Factorial!Example 8.1.6: 5 people on a committee need to be lined up for a photo. In how many ways can this be done?

Factorial!Example 8.1.7: 42 cars are in a race. How many possible finishing orders can there be?

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Permutations• Let’s consider the last example. What if we only cared

about the possible top 5 finishing orders?

Permutations

Go to math and arrow over to

PRB.

PermutationsExample 8.1.8: You have 6 books and want to place 4 of them on a shelf. In how many ways can you organize the books?

PermutationsExample 8.1.9: There are 15 people on a committee. How many ways can they select a president, vice president, and treasurer?

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PermutationsExample 8.1.10: A television talk show will include 4 men and 5 women as panelists.

(a) In how many ways can the panelists be seated?

(b) In many ways can they be seated if they need to alternate?

(c) In how many ways can they be seated if the men and women must be seated together?

Repetition in Permutations• Suppose you are asked to arrange some objects but the

objects are indistinguishable.• For example, arranging the numbers 1, 2, 2, 1, 1.▫ You cannot tell the ones apart or the twos apart.

• If you have to arrange objects where of them repeat, of them repeat, of them repeat, … Then you can

calculate the number of arrangements as!⋯

Permutations (Repetitions)Example 8.1.11: In how many ways can you arrange the letters of the word “Mississippi”?

Permutations (Repetitions)• Recall when were taking marbles out of the bag and I kept

saying, “in that order.”• Now we don’t need to worry about the order.

Example 8.1.12: A bag contains 5 blue, 4 black, 6 yellow, and 5 white marbles. If two blues and a black marble were drawn, how many ways can they be arranged?

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Permutations (Repetitions)Example 8.1.13: You selected a full house with 3 tens and 2 aces. In how many ways could the cards have been drawn?

License PlatesExample 8.1.14: For many years, the State of California used3 letters followed by 3 digits on its automobile license plates.

(a) How many different license plates are possible with this arrangement?

(b) When the state ran out of new numbers, the order was reversed to 3 digits followed by 3 letters. How many new license plate numbers were then possible?

(c) Several years ago. the numbers described in b were also used up. The stale then issued plates with I letter followed by 3 digits and then 3 letters. How many new license plate numbers will this provide?

Combinations

Combination

124 495

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Combination versus Permutation• A permutation is when order matters.▫ Lining people up.▫ Selecting people from a group and assigning them positions.

• A combination is when order does not matter.▫ Selecting cards▫ Picking people for a team.

Combinations vs. PermutationsExample 8.2.1: For each problem, tell whether permutations or combinations should be used to solve the problem.

(a) How many 4-digit code numbers arc possible if no digits are repeated?

(b) A sample of 3 light bulbs is randomly selected from a batch of 5. How many different samples are possible?

(c) In a baseball conference with 8 teams, how many games must be played so that each team plays every other team exactly once?

(d) In how many ways can 4 patients be assigned to 6 different hospital rooms so that each patient has a private room?

Locker “Combination”• If you have a locker “combination” does the order

matter?

CombinationsExample 8.2.2: There are 15 people in a class. How many ways can 4 of them be selected to work on a project together?

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CombinationsExample 8.2.3: There are 52 cards in a standard deck of cards. In how many ways can you select 5 cards?

CombinationsExample 8.2.4: There are 20 people in Club West. Five students must be selected to work on an activity.

(a) How many ways can this be done?

(b) How many ways can this be done if Joe must be in the group?

(c) What if the question was change to at most 3 people need to selected for the group?

CombinationsExample 8.2.5: In how many ways can you select a full house with tens over aces?

CombinationsExample 8.2.6: When selecting a five card hand:

(a) How many possible hands are there with two hearts?

(b) How many possible hands are there with all face cards?

(c) How many possible hands are there where all 5 cards are of the same suite?

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CombinationsExample 8.2.7: How many possible full houses are there?

CombinationsExample 8.2.8: The senate contains 51 democrats, 47 republicans, and 1 independent. How many ways can a committee be formed if it must contain 3 democrats and 2 republicans?

Probability Applications of Counting Principles

Probability• Recall that probability can be calculated as

• For more complicated situations we can use the multiplication principle, combinations, and permutations to calculate probability.

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CombinationsExample 8.3.1: There are 20 people in Club West, 12 of which are women. Five students must be selected to work on an activity. If the people were randomly chosen,

(a) What is the probability of there being 3 women and 2 men?

(b) What is the probability of there being all women?

CombinationsExample 8.3.2: There are 52 cards in a standard deck of cards. What is the probability of getting four of a kind?

Combinations:

Using the Multiplication Principle:

CombinationsExample 8.3.3: What is the probability of being dealt a full house?

Combinations:

Using the Multiplication Principle:

CombinationsExample 8.3.4: The senate contains 51 democrats, 47 republicans, and 1 independent. A committee of 10 must be formed. If the people were randomly chosen,

(a) What is the probability of there being 3 democrats, 6 republicans, and 1 independent?

(b) What is the probability of there being 9 democrats, and 1 republican?

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CombinationsExample 8.3.5: Find the probability of each.

(a) A 5-card hand with two hearts?

(b) A 5-card hand with all face cards?

(c) How many possible hands are there where all 5 cards are of the same suite?

We don’t need no stinkn’ order!• Remember when we did probability before and I kept

saying, “in that order.”

• We don’t need it any more.

ProbabilityExample 8.3.6: A bag of marbles contains 3 red, 5 blue, 7 yellow, and 5 green marbles. If 3 are drawn, what is the probability of

(a) Drawing 3 marbles and they are yellow, green, and blue?

(b) Drawing 1 or more green marbles?

ProbabilityExample 8.3.7: The Environmental Protection Agency is considering inspecting 6 plants for environmental compliance: 3 in Chicago, 2 in Los Angeles, and 1 in New York. Due to a lack of inspectors, they decide to inspect 2 plants selected at random, 1 this month and 1 next month, with each plant equally likely to be selected, but no plant is selected twice. What is the probability that 1 Chicago plant and 1 Los Angeles plant are selected?

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ProbabilityExample 8.3.8: From a group of 50 accountants at a firm 5 are selected to write a review of management policies.

(a) How many possible groups of 5 can be made?

(b) What is the probability that Jessica Smith, one of the accountants, is selected?

ProbabilityExample 8.3.9: During the 1988 college football season, the Big Eight conference ended the season with a perfect progression.

(a) How many games did the 8 teams play?

(b) Assuming no ties, how many differentoutcomes are there for all the games together?

(c) In how many ways could the eight teams endin a perfect progression?

(d) Assuming each team had an equal chance of winning, what is the probability of a perfect progression.

Binomial Probability

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Definition

• Flipping a coin.• Do a behavior, don’t do a behavior.• Faulty, not faulty.

Binomial Probability Formula

ProbabilityExample 8.4.1: You flip a coin 10 times. What is the probability of getting 4 heads?

ProbabilityExample 8.4.2: Over a long period of time it has been observed that a given rifleman can hit a target on a single trial with probability equal to .8. Suppose that he has four shots at a target:

(a) What is the probability that he will hit it twice?

(b) What is the probability that he will hit it at least twice?

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ProbabilityExample 8.4.3: A bag contains 4 red, 5 green, 7 blue, and 4 white marbles.

(a) What is the probability of drawing 10 marbles and three of them are blue?

(b) What is the probability of drawing 10 marbles and at least four of them are red?

ProbabilityExample 8.4.4: A company produces light bulbs. It has found that 1 out of every 10,000 is faulty. If a sample of 30 light bulbs is taken what it the probability that at least one is faulty?