HSC Mathematics Revision: Permutations and Combinations Extension 1
27 Questions
1 Views

HSC Mathematics Revision: Permutations and Combinations Extension 1

Created by
@LeadingVirginiaBeach

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What is the formula for the number of permutations of n objects taken r at a time?

  • n! / (n – r)!r!
  • n! × (n – r)!
  • n × r!
  • n! / (n + r)! (correct)

    • In how many ways can 4 speakers be arranged in a row for a photo?

  • 4 × 3!
  • 4!
  • 4 × 3 × 2 × 1! (correct)
  • 4 × 3 × 2!

    • How many ways can 5 boys and 4 girls be arranged on a bench if boys and girls are in separate groups?

  • 5! × 4! × 2
  • 5! × 4!
  • 5! × 4! × 2
  • 5! × 4! × 2! (correct)

    • How many ways can 5 boys and 4 girls be arranged on a bench if Anne and Jim wish to stay together?

    <p>2 × 8!</p> Signup and view all the answers

    What is the number of ways to arrange 5 objects in a circle?

    <p>(5-1)!</p> Signup and view all the answers

    How many 4-digit numbers greater than 4000 can be formed from the digits 2, 3, 4, 5, and 6?

    <p>3P1 × 4P3</p> Signup and view all the answers

    What is the number of ways 6 men and 6 women can sit at a round table if men and women alternate?

    <p>5! × 6!</p> Signup and view all the answers

    How many 4-digit even numbers can be formed from the digits 2, 3, 4, 5, and 6?

    <p>4P3 × 3P1</p> Signup and view all the answers

    What is the number of ways 6 men and 6 women can sit at a round table if there are no restrictions?

    <p>11!</p> Signup and view all the answers

    How many 5-digit numbers greater than 4000 can be formed from the digits 2, 3, 4, 5, and 6?

    <p>5P5</p> Signup and view all the answers

    What is the total possible number of hands in a poker game where 5 cards are dealt from a regular pack of 52 cards?

    <p>52C5</p> Signup and view all the answers

    How many hands of poker have 4 Kings?

    <p>C4 × 48C1</p> Signup and view all the answers

    How many ways can 4 Maths books be arranged from 6 different Maths books?

    <p>6P4</p> Signup and view all the answers

    What is the number of ways to choose 3 Aces and 2 Kings from a deck of 52 cards?

    <p>C3 × 4C2</p> Signup and view all the answers

    What is the number of ways to arrange 7 books on a shelf where 4 Maths books are chosen from 6 different Maths books and 3 English books are chosen from 5 different English books, with no restrictions?

    <p>6C4 × 5C3 × 7!</p> Signup and view all the answers

    If a bookshelf has 8 shelves and each shelf can hold 5 books, how many ways can the books be arranged on the shelves?

    <p>5 × 5 × 5 × 5 × 5 × 5 × 5 × 5</p> Signup and view all the answers

    A die is rolled 4 times. What is the number of possible outcomes?

    <p>6^4</p> Signup and view all the answers

    A license plate consists of 3 letters and 2 digits. How many possible license plates are there?

    <p>26 × 26 × 26 × 10 × 10</p> Signup and view all the answers

    If 5 people are to be arranged in a row, what is the number of possible arrangements?

    <p>5 × 4 × 3 × 2 × 1</p> Signup and view all the answers

    If a bag contains 6 balls and 3 balls are drawn at random, what is the number of possible outcomes?

    <p>6 × 5 × 4</p> Signup and view all the answers

    If a word consists of 4 letters and each letter can be chosen from 26 possible letters, what is the number of possible words?

    <p>26 × 26 × 26 × 26</p> Signup and view all the answers

    If 4 Maths books are selected from 6 different Maths books and 3 English books are chosen from 5 different English books, how many ways can the seven books be arranged on a shelf if a Maths book is at the beginning of the shelf?

    <p>6 × 5C3 × 5C3 × 6!</p> Signup and view all the answers

    If 4 Maths books are selected from 6 different Maths books and 3 English books are chosen from 5 different English books, how many ways can the seven books be arranged on a shelf if Maths and English books alternate?

    <p>6P4 × 5P3</p> Signup and view all the answers

    If 4 Maths books are selected from 6 different Maths books and 3 English books are chosen from 5 different English books, how many ways can the seven books be arranged on a shelf if a Maths book is at the beginning and an English book is in the middle of the shelf?

    <p>6 × 5 × 5C3 × 4C2 × 5!</p> Signup and view all the answers

    How many different 8 letter words are possible using the letters of the word SYLLABUS?

    <p>10 080</p> Signup and view all the answers

    If a word is chosen at random from the letters of the word SYLLABUS, what is the probability that the word contains the two S’s together?

    <p>2520 / 10 080</p> Signup and view all the answers

    What is the formula to find the number of ways to arrange the seven books on a shelf if a Maths book is at the beginning of the shelf?

    <p>6 × 5C3 × 5C3 × 6!</p> Signup and view all the answers

    Study Notes

    Permutations and Combinations

    • The multiplication rule states that if one event can occur in m ways, a second event in n ways, and a third event in r ways, then the three events can occur in m × n × r ways.
    • Repetition of an event:
      • If one event with n outcomes occurs r times with repetition allowed, then the number of ordered arrangements is n^r.
      • Example: Rolling a die r times, the number of arrangements is 6^r.
    • Factorial representation:
      • n! = n(n – 1)(n – 2)………..3 × 2 × 1
      • Example: 5! = 5 × 4 × 3 × 2 × 1
      • Note: 0! = 1
    • Arrangements or permutations:
      • The number of permutations of n objects taken r at a time is given by nPr = n! / (n – r)!.
      • Example: 4P4 = 4! and 4P2 = 12
    • Permutations with restrictions:
      • Example: 5 boys and 4 girls can be arranged on a bench in 5! × 4! ways if there are no restrictions.
      • If boys and girls alternate, the number of arrangements is 5! × 4!.
      • If boys and girls are in separate groups, the number of arrangements is 2 × 5! × 4!.
      • If Anne and Jim wish to stay together, the number of arrangements is 2 × 8!.
    • Further permutations and combinations:
      • Example: 4 Maths books are selected from 6 different Maths books, and 3 English books are chosen from 5 different English books. The number of ways to arrange the 7 books on a shelf is 6C4 × 5C3 × 7!.
      • If a Maths book is at the beginning of the shelf, the number of arrangements is 6 × 5C3 × 5C3 × 6!.
      • If Maths and English books alternate, the number of arrangements is 6P4 × 5P3.
      • If a Maths book is at the beginning and an English book is in the middle of the shelf, the number of arrangements is 6 × 5 × 5C3 × 4C2 × 5!.
    • Circular arrangements:
      • The number of ways to arrange n objects in a circle is (n – 1)!.
      • Example: 6 men and 6 women sit at a round table. If there are no restrictions, the number of ways they can sit is 11!.
      • If men and women alternate, the number of arrangements is 5! × 6!.
      • If Ted and Carol must sit together, the number of arrangements is 2! × 10!.
      • If Bob, Ted, and Carol must sit together, the number of arrangements is 3! × 9!.
    • Combinations:
      • The number of combinations of n objects taken r at a time is given by nCr = n! / (r! × (n – r)!).
      • Example: 52C5 is the total possible number of hands in a game of poker.
      • Example: The number of hands with 4 Kings is C4 × 48C1.
      • Example: The number of hands with 2 Clubs and 3 Hearts is C2 × 13C3.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Description

    Test your understanding of the multiplication rule and repetition of events in permutations and combinations. This quiz is designed for HSC mathematics students revising for their exams.

    More Like This

    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser