11.1 Permutations and Combinations 1 Chapter 11: Probability and Statistics 11.1 Permutations and Combinations

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    Chapter 11: Probability and Statistics

    11.1 Permutations and Combinations The Fundamental Counting Principle allow us to count large numbers of possibilities quickly.

    You can extend the idea to any number of choices. Example 1: A college offers 3 different English courses, 5 different math course, 2 different art courses, and 4 different history courses. In how many ways could a student choose 1 of each type of course? Frequently with counting problems you will use combinations of letters and digits. There are ____ different digits possible: There are ____ letters in the alphabet.

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    Permutations A permutation is an _______________________________ of items in a particular order. Example 2: How many ways can 5 Scrabble tiles be placed in a row? Any time you have a number in the form N x (N-1) x (N-2) x . . . x 3 x 2 x 1 you can write it as N! (read “N Factorial”) Example 3 a) How many ways can 10 different b) How many ways can you arrange textbooks be arranged on a shelf? 8 different shirts on hangers in your closet? Practice with Factorials Example 4: Calculate the following

    A) 5! B) 15! C) 12!

    10! D)

    8!

    4!

    E) 5(4!) F) 10!

    6!4! G)

    11!

    7!2!

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    Example 5: Ten students are in a race. How many ways can 10 runners finish 1st, 2nd, and 3rd? (no ties allowed) Method 1: Fundamental Counting Principle Method 2: nPr Formula Most scientific calculators have and nPr function to compute the number of permutations of n objects taken r at a time. TI 83 Type in “n” value Press “MATH” button. Arrow over to PRB Choose option #2 Type in “r” value Hit enter n P r = 10 P 3

    Example 6: Calculate the following a) 12P8 b) 20P5

    COUNTING THE NUMBER OF PERMUTATIONS Sometimes we want to arrange items from a set, but we don’t want to use ALL the items

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    Example 7: The drama club has 20 members. In how many ways cans you choose a president, a vice president, and a secretary? Example 8: A teacher is making a seating chart. In how many ways can she assign 35 students to the eight chairs in the front row? Example 9: A child is playing with 4 Fisher Price “Little People”. In how many ways can she line them up in a row?

    Permutations with Repetition Example 10: How many ways can you arrange the letters of the word “PARTY” ? Example 11: How many ways can you arrange the letters of the word “HAPPY”? Example 12: How many ways can you arrange the letters of the word “MISSISSIPPI” ?

    Repetition? If you are assigning position to things and some of the items are identical… Example: N items x are the same y are the same

    #arrangements = !

    ! !

    N

    x y

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    Example 13: How many ways can you make a stack of 5 blocks, 3 of which are red and 2 of which are blue?

    Combinations A combination is a _____________________ of items in which order doesn’t necessarily matter.

    Example 1: Five men are nominated for the Academy Award for “Best Actor.” Choose your two favorite actors: ____________________________ ____________________________ The ORDER you write down your answers doesn’t matter since you were not asked to RANK the actors. If you write down: “Christian & Chiwetel” and your neighbor writes down: “Chiwetel & Christian”, you would both claim to have the same favorite actors. Example 2: “How many ways can you select three tiles from a set of five?”

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    For a TI83, the nCr command is located right under the nPr command in the Math -> PRB menu

    Example 3: Calculate the following a) 15C3 b) 12C9 c) 3(8C5)

    Example 4: For each problem, consider whether you should use counting principle, permutations, or combinations to solve. Then find the indicated number of things. a) A chemistry teacher divides his class into eight groups. Each group submits one drawing of the molecular structure of water. In how many different ways can he select 4 of the drawings to display? b) You will draw winners from a total of 25 tickets in a raffle. The first ticket wins $100. The second ticket wins $50. The third ticket wins $10. In how many different ways can you draw the three winning tickets? c) Donna Noble is buying wedding clothes. She has it narrowed down to 5 dresses, 2 veils, and 8 pairs of shoes. How many possible outfits can she make by choosing 1 dress, 1 veil, and 1 pair of shoes?

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    (Example 4 continued)

    d) How many “words” can you make from the letters of the word “BASEBALL”?

    e) How many blends of three spices can you make from a selection of 8 spices from the spice rack? f) A company is trying to hire office workers from 30 qualified applicants. How many ways can they schedule interviews for 5 people tomorrow? How many ways can they select 3 people to hire? g) How many ways can you arrange 5 books on a shelf? h) A combination lock has 3 digits. How many combinations are possible if you may only use the digits 0 to 9? Food for thought: Is it appropriate to call a combination lock a combination lock?

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    11.2 Probability Probability is how _________________ an event is to occur. The probability of an event is measured somewhere between 0 and 1. P(impossible) = ______ P(certain) = ______ Many times probabilities are expressed as fractions or as percentages. Actually collecting data: Example 1: Of the 60 vehicles in the teachers’ parking lot today, 15 are pickup trucks. What is the experimental probability that a vehicle in the lot is a pickup truck? What is the probability that a vehicle is NOT a pickup truck? Example 2: A softball player got a hit in 20 of her last 50 times at bat. What is the experimental probability that she will get a hit in her next at bat? A simulation is a model of an event. You can use a simulation to find the experimental probability of an event. Example: What is the probability of passing a 10 question multiple choice quiz with 5 answers choice for each question. Theoretical: P(guess correct) = 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9  Assign ____ % “correct” Randomly decide where to start: Say, Row 5, column 3 Write down 10 numbers (for the 10 questions): Circle “correct”

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    The sample space is the set of all possible outcomes of an experiment. When each outcome has the same chance of occurring, the outcomes are called equally likely outcomes. Example 3: Given a standard number cube, what is the theoretical probability of each event? a) P(even) b) P(7) c) P(even or multiple of 3) Example 4: Given a standard deck of 52 playing cards, what is the theoretical probability of each event? Note: 13 Cards: Ace, King, Queen, Jack, 10,9,8,7,6,5,4,3,2 4 Suits: Black Suits – Clubs, Spades Red Suits – Diamonds, Hearts a) P(King) b) P(Face Card) c) P(heart or a 10) Example 5: You open a bag of jawbreakers and find 15 red, 10 orange, 8 green, and 7 purple. What is the theoretical probability of each event? a) P (red or orange) b) P(not green) c) P(red or not orange)

    Rolling a number cube:

    Sample Space:

    Equally likely?

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    Combinatorics When you are simultaneously choosing several things at once, you may need to use combinations to determine the probability of an event. Example 6: A student has a personal library of 30 zombie movies and 10 action movies. If the student randomly grabs 6 movies to take on vacation, what are the following theoretical probabilities? Sample Space: a) P(all zombie movies) b) P(all action movies) c) P(exactly 2 zombie movies) d) P(exactly 4 zombie movies) Example 7: What is the theoretical probability of being dealt the following 5 card hands from a standard 52-card deck? Sample Space: a) P(5 diamonds) b) P (all 2’s or 3’s) c) P(exactly 2 kings) c) P(exactly 3 hearts)

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    11.3 Probability of Multiple Events Consider the following: I have a stack of 10 cards each with a different number from 1 to 10. 1,2,3,4,5,6,7,8,9,10 Simple Probability (1 item, 1 time)

     You draw 1 card from the stack. What is the probability that the card is an even number? Combinatorics (several items at the same time)

     You choose 3 cards from the stack. What is the probability that all three cards are even?

     You choose 3 cards from the stack. What is the probability of getting exactly 2 odd-numbered cards?

    Multiple Events (1 item, several trials or different items)

     You roll a standard number cube and flip a coin. What is the probability of getting a even number and heads?

     You flip a coin 6 times in a row. What is the probability of getting all heads?