Gr 12 Mathematics: Ch 9.5 Application to counting Problems
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Questions and Answers

What does the notation n! (read as 'n factorial') represent?

  • The product of all positive integers excluding n
  • The product of all positive integers up to n (correct)
  • The sum of all positive integers up to n
  • The product of all positive integers greater than n

    • What is the value of 0!?

  • 1 (correct)
  • undefined
  • -1
  • 0

    • What is the total number of possible arrangements of n different objects?

  • n^2
  • n^3
  • n! (correct)
  • 2^n

    • What is the value of n! / (n-1)! for any positive integer n?

    <p>n</p> Signup and view all the answers

    How many choices are available for the first place in a 400 m race with 8 athletes?

    <p>8</p> Signup and view all the answers

    What is the purpose of factorial notation in counting problems?

    <p>To simplify expressions involving permutations and combinations</p> Signup and view all the answers

    What is the number of different ways in which the first three places can be arranged in a 400 m race with 8 athletes?

    <p>6 × 7 × 8</p> Signup and view all the answers

    What type of problems frequently use factorial notation?

    <p>Counting problems</p> Signup and view all the answers

    What is the number of different ways to arrange the first three places in a competition with 8 contestants?

    <p>336</p> Signup and view all the answers

    If seven boys must sit in a row, and the youngest and oldest must sit together, how many total arrangements are possible?

    <p>1440</p> Signup and view all the answers

    If seven boys must sit in a row, but the youngest and oldest must not sit together, how many total arrangements are possible?

    <p>3600</p> Signup and view all the answers

    What is the formula to find the total number of unique arrangements of letters in a word with repeated letters?

    <p>n! / (k! × m!)</p> Signup and view all the answers

    What is the probability of an event in terms of counting principles?

    <p>Number of favourable outcomes divided by the total number of outcomes</p> Signup and view all the answers

    How many arrangements are there for a word with 7 letters, where 2 letters are repeated?

    <p>7! / (2! × 2!)</p> Signup and view all the answers

    What is the purpose of counting principles in probability?

    <p>To calculate the probability of an event</p> Signup and view all the answers

    How do you simplify an arrangement problem with constraints?

    <p>By treating the constrained pair as a single entity</p> Signup and view all the answers

    What is the value of 5! + 4!?

    <p>126</p> Signup and view all the answers

    How many different ways can 5 books be arranged on a shelf?

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

    If 6 people are attending a dinner party, how many ways can they be seated around a circular table?

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

    What is the value of (\frac{6!}{3!})?

    <p>36</p> Signup and view all the answers

    In a group of 9 students, how many ways can 3 students be chosen to represent the class?

    <p>9(\choose{3})</p> Signup and view all the answers

    What is the value of (\frac{4!}{2!})?

    <p>8</p> Signup and view all the answers

    In a game of chess, there are 16 possible moves for white's pawn. If black responds with one of 15 possible moves, how many different outcomes are possible for the first two moves?

    <p>16 * 15</p> Signup and view all the answers

    A code consists of 5 digits, where each digit can be any of the numbers 0-9. How many possible codes are there?

    <p>10^5</p> Signup and view all the answers

    In an arrangement problem with constraints, what is the general strategy to simplify the problem?

    <p>Combine all the constrained elements into one entity</p> Signup and view all the answers

    What is the formula to find the total number of unique arrangements of letters in a word with repeated letters?

    <p>(n!)/((n-k)! × k!)</p> Signup and view all the answers

    In a probability problem, what is the formula to calculate the probability of an event?

    <p>Number of favourable outcomes / Total number of possible outcomes</p> Signup and view all the answers

    In an arrangement problem, what happens to the number of arrangements when a constraint is applied?

    <p>The number of arrangements decreases</p> Signup and view all the answers

    What is the purpose of counting principles in probability?

    <p>To calculate the probability of an event</p> Signup and view all the answers

    How do you calculate the total number of arrangements when some objects must not sit together?

    <p>Calculate the total number of arrangements without constraints and subtract the number of arrangements where they sit together</p> Signup and view all the answers

    What is the formula to find the total number of arrangements when some objects are treated as a single entity?

    <p>(n-1)! × 2!</p> Signup and view all the answers

    What is the purpose of treating constrained elements as a single entity in an arrangement problem?

    <p>To simplify the problem</p> Signup and view all the answers

    In an arrangement problem with constraints, what happens to the number of arrangements when the constraint is removed?

    <p>The number of arrangements increases by the factorial of the number of entities.</p> Signup and view all the answers

    What is the probability that a random arrangement of the letters in the word 'BANANA' starts and ends with the letter 'A'?

    <p>1/4</p> Signup and view all the answers

    A group of 8 friends want to sit in a row, but 2 friends, Alice and Bob, refuse to sit together. How many arrangements are possible?

    <p>8! - 7!*2!</p> Signup and view all the answers

    What is the total number of unique arrangements of the letters in the word 'STATISTICS'?

    <p>10! / (2!*2!*2!*4!)</p> Signup and view all the answers

    For a set of 10 different objects, which of the following is NOT a correct way to calculate the total number of possible arrangements?

    <p>(10 - 1)!</p> Signup and view all the answers

    A code consists of 7 digits, where each digit can be any of the numbers 0-9. How many possible codes are there if the first and last digits must be the same?

    <p>10^6</p> Signup and view all the answers

    What is the relationship between the number of possible arrangements of n objects and the number of possible arrangements of (n - 1) objects?

    <p>The number of possible arrangements of n objects is n times the number of possible arrangements of (n - 1) objects.</p> Signup and view all the answers

    A group of 12 people are attending a dinner party, and they can be seated in any of the 12 chairs around a circular table. How many possible seating arrangements are there?

    <p>(12 - 1)!</p> Signup and view all the answers

    What is the probability that a random arrangement of 7 boys sitting in a row has the youngest and oldest sitting together?

    <p>1440 / 5040</p> Signup and view all the answers

    What is the purpose of using factorial notation in counting problems?

    <p>All of the above</p> Signup and view all the answers

    In how many ways can 7 different books be arranged on a shelf if 2 books are to be placed together?

    <p>6! × 2!</p> Signup and view all the answers

    Five different books are to be arranged on a shelf. How many possible arrangements are there?

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

    What is the total number of arrangements of 9 people sitting in a row, where 3 friends refuse to sit together?

    <p>9! - 7!*3!</p> Signup and view all the answers

    Which of the following is a correct application of factorial notation?

    <p>Finding the number of ways to arrange 5 different objects in a row.</p> Signup and view all the answers

    A box contains 8 different toys, and 3 of them are to be arranged in a row. How many possible arrangements are there?

    <p>8 × 7 × 6</p> Signup and view all the answers

    What is the value of (\frac{8!}{5!})?

    <p>8 × 7 × 6</p> Signup and view all the answers

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