HSC Mathematics Revision: Permutations and Combinations Extension 1
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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.

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    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.

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