Distinguishable Permutations and Combinations
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Questions and Answers

In an election with 12 candidates, how many ways can a voter select 6 councilors?

  • 924 (correct)
  • 462
  • 132
  • 665,280

    • A school photography club has 9 boys and 6 girls. How many ways can a committee of 5 students be selected from this club?

  • 150
  • 792
  • 3003 (correct)
  • 15

    • What is the formula for the probability of event A or event B occurring when A and B are not mutually exclusive?

  • $P(A \text{ or } B) = P(A) + P(B)$
  • $P(A \text{ or } B) = P(A) + P(B) + P(A \text{ and } B)$
  • $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$ (correct)
  • $P(A \text{ or } B) = P(A) * P(B)$

    • A bag contains 4 blue, 3 white, and 5 red marbles. Two marbles are drawn without replacement. What is the probability the first marble is red and the second is blue?

    <p>$5/33$ (C)</p> Signup and view all the answers

    You flip a coin and roll a six-sided die. What is the probability of getting a tail on the coin and an even number on the die?

    <p>$1/4$ (B)</p> Signup and view all the answers

    What term describes different arrangements of objects where some are identical?

    <p>Distinguishable Permutation (A)</p> Signup and view all the answers

    When the order of objects is considered, what is an arrangement of those objects called?

    <p>Permutation (D)</p> Signup and view all the answers

    Which mathematical expression is equivalent to P(10, 5)?

    <p>10P5 (A)</p> Signup and view all the answers

    What is the mathematical expression that represents the combination of n objects taken r at a time?

    <p>nCr (D)</p> Signup and view all the answers

    Which of the following scenarios exemplifies a combination?

    <p>Choosing a subset of a set (C)</p> Signup and view all the answers

    What mathematical expression represents the permutation of n objects taken r at a time?

    <p>nPr (D)</p> Signup and view all the answers

    If A = {1, 3, 5, 7} and B = {2, 3, 4, 5}, what is the intersection of A and B (A ∩ B)?

    <p>{3, 5} (A)</p> Signup and view all the answers

    Consider two events: A = {2, 4, 6, 8} and B = {3, 5, 6, 7, 8}. Which set correctly represents (A ∩ B)?

    <p>{6, 8} (D)</p> Signup and view all the answers

    Flashcards

    Distinguishable Permutation

    Arrangements of objects where some are identical.

    Permutation

    An arrangement of objects where order matters.

    Combination

    Choosing a subset of a set without regard to order.

    Mathematical Notation for Combination

    The expression that illustrates combinations is nCr.

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    Mathematical Notation for Permutation

    The expression for permutations is nPr.

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    Intersection of Events

    The notation for intersection of two events A and B is A ∩ B.

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    Finding n in Permutation

    If P(n, 4) = 5,040, then n must be 8.

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    Count of Distinguishable Permutations

    Total distinct arrangements for the word EDUCATED considering duplicates.

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    Probability of face cards

    The likelihood that both drawn cards are face cards from a deck.

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    Mutually exclusive events formula

    P(A or B) = P(A) + P(B), used for events that cannot happen simultaneously.

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    Not mutually exclusive events formula

    P(A or B) = P(A) + P(B) - P(A and B), for events that can occur together.

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    Venn diagram representation

    A visual tool used to show the union and intersection of two sets.

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    Probability of dog or cat household

    The chances that a randomly selected household owns a dog or cat.

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    Study Notes

    Distinguishable Permutations

    • A distinguishable permutation is an arrangement of objects where some objects are identical.

    Permutation

    • A permutation is an arrangement of objects where the order of the objects is considered.
    • The notation for a permutation of n objects taken r at a time is P(n, r).

    Combination

    • A combination is a subset of a set where the order of the objects is not considered.
    • The notation for a combination of n objects taken r at a time is nCr.

    Mathematical Expression for Combination

    • The mathematical expression for choosing r objects from a set of n objects is nCr.

    Permutations of Letters in "EDUCATED"

    • The number of distinguishable permutations of the letters in the word "EDUCATED" is calculated by considering the repeated letters.

    Intersection of Events

    • If A and B are two events, their intersection (A ∩ B) is the set of all outcomes that are in both A and B.

    Combination (8,8)

    • C(8,8) = 1

    Value of 'n' in a Permutation

    • If P(n, 4) = 5,040, then n = \210

    Combination of 7 Objects Taken 3 at a Time

    • The combination of 7 objects taken 3 at a time is 7C3, which equals 35.

    Permutation of 6 Objects Taken 2 at a Time

    • The permutation of 6 taken 2 at a time is P(6, 2), which is equal to 30.

    Probability of Intersection of Events

    • The provided examples show how to determine the intersection of two sets given the possible outcomes.
    • For a given example, A = {2, 4, 6, 8} and B = {3, 5, 6, 7, 8}, A ∩ B= {6, 8}

    Students in English & Chemistry classes

    • Out of 50 students, 22 are taking English, 25 are taking Chemistry, and 10 are taking both. The number of students in neither class is 18.

    Selecting Face Cards from a Deck

    • The number of ways to select seven face cards from a standard deck of cards.

    Arranging Singers in a Competition

    • The number of ways to arrange the first five singers from a group of 10 contestants in a singing competition, considering the order.

    Probability of Facades cards

    • The problem describes drawing two cards from a deck (with replacement) and determines the probability of both being face cards.

    Committee of Students (Juniors & Seniors)

    • The number of ways to choose a committee of 7 students from 9 juniors and 9 seniors, with 4 seniors in the committee.

    Student Committee Selection

    • In a photography club with 9 boys and 6 girls, the number of ways to select a committee made up of 5 students.

    Councilors' Selection

    • Given 12 candidates, the number of possible ways a voter can select 6 councilors for a local election.

    Formula for Not Mutually Exclusive/Inclusive Events

    • The formula for not mutually exclusive events is given (the calculations aren't shown)

    Mutually Exclusive Event Formula

    • Given is the formula describing mutually exclusive events (calculations unclear)

    Venn Diagram of Union & Intersection of Events

    • A Venn diagram is to be drawn demonstrating the union and intersection of two sets, {1,3,5}, and {2,3,5}

    Probability of Dog or Cat Ownership

    • The probability of a household randomly selected owning a dog or a cat (or both) is defined for a given set of data.

    Probability of Head or Tail in Coin Flips

    • The probability of getting a head or a tail when flipping a coin.

    Probability of Red Then Blue Marbles

    • The probability of drawing a red marble, then a blue marble from a bag with different-colored marbles, without replacement.

    Probability of Drawing Three White Balls

    • The problem gives the details regarding drawing three white balls in succession (with replacement) from a jar containing white and black balls.

    Probability of Twenty and Fifty Peso Bills

    • The probability of picking a twenty peso bill and then a fifty peso bill, without replacement, from a wallet containing both.

    Probability of Tail and Even Number

    • The probability of getting a tail when flipping a coin, and an even number when rolling a standard die.

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    Description

    This quiz covers the concepts of distinguishable permutations, permutations, and combinations, including their definitions and mathematical expressions. Explore practical examples like the permutations of the letters in 'EDUCATED' and the intersection of events to solidify your understanding.

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