Permutation and combination - (14.1-14.3)

Understand the addition rule and multiplication rule in the counting principle

  • Theory
  • Examples
    a) In S5A, students must choose to study at least one subject between History and Geography.
    There are 38 students in the class, 30 of them study History and 27 of them study Geography.
    How many students in S5A study both History and Geography?

    b) In S5A, S5B and S5C, there are 8 students, 10 students and 13 students respectively who are
    qualified to join a competition. If 3 qualified students are selected, how many ways are there to
    select one student from each class?
  • Solutions
    a) Let x be the number of students in S5A study both History and Geography.
    \(\begin{array}{c}38 = 30 + 27 - x\\x = {19^{}}\end{array}\)
    Therefore, 19 students in S5A study both History and Geography.

    b)Number of ways
    \(\begin{array}{l} = 8 \times 10 \times 13\\ = {\underline{\underline {1{\rm{ }}040}} ^{}}\end{array}\)

Understand the concept and notation of permutation

  • Theory
  • Examples
    Evaluate each of the following. (Express your answers in terms of n.)
    a) \(P_2^{n + 2}\)
    b) \(P_n^{n + 2}\)
    c) \(P_3^{n + 1} - P_3^n\)
    d) \(\frac{{P_3^{n + 2}}}{{P_3^{n + 1}}}\)
  • Solutions
    \(\begin{array}{c} a)\qquad P_2^{n + 2} = \underline{\underline {(n + 2)(n + 1)}} \end{array}\)

    \(\begin{array}{c} b)\qquad P_n^{n + 2} = \frac{{(n + 2)!}}{{(n + 2 - n)!}} = {\underline{\underline {\frac{{(n + 2)!}}{2}}} ^{}}\end{array}\)

    \(\begin{array}{c}c)\qquad P_3^{n + 1} - P_3^n = (n + 1)(n)(n - 1) - n(n - 1)(n - 2)\\ = n(n - 1){(n + 1 - n + 2)^{}}\\ = {\underline{\underline {3n(n - 1)}} ^{}}\end{array}\)

    \(\begin{array}{c}d)\qquad \frac{{P_3^{n + 2}}}{{P_3^{n + 1}}} = \frac{{(n + 2)(n + 1)(n)}}{{(n + 1)(n)(n - 1)}}\\ = \underline{\underline {\frac{{n + 2}}{{n - 1}}}} \end{array}\)

Solve problems on the permutation of distinct objects without repetition

  • Theory
  • Examples
    A five-digit number is formed by 5 digits selecting from 9 numbers from 1 to 9.
    a) If the units digit is 2, how many different five-digit numbers can be formed?
    b) If the units digit is an even number, how many different five-digit numbers can be formed?
  • Solutions

    a)Number of five-digit numbers\(\begin{array}{l} \\ = P_4^8 \\ = \underline{\underline {1{\rm{ }}680}} \end{array}\)

    b)Number of five-digit numbers\(\begin{array}{l} \\ = P_4^8 + P_4^8 + P_4^8 + P_4^8\\ = \underline{\underline {6{\rm{ }}720}} \end{array}\)