Permutation and combination - (14.4-14.5)

Understand the concept and notation of combination

  • Theory
  • Examples
    Evaluate each of the following.
    a) \(C_1^3 + C_2^3 + C_3^3\)
    b) \(C_2^4 \times C_2^5\)
    c) \(C_5^9 \div C_7^9\)
  • Solutions
    \(\begin{array}{c}a)\qquad C_1^3 + C_2^3 + C_3^3 = \frac{{3{\rm{ }}!}}{{2{\rm{ }}!{\rm{ }} \times 1{\rm{ }}!}} + \frac{{3{\rm{ }}!}}{{1{\rm{ }}!{\rm{ }} \times 2{\rm{ }}!}} + \frac{{3{\rm{ }}!}}{{0!{\rm{ }} \times 3{\rm{ }}!}} \\ = \frac{{3 \times 2}}{{2 \times 1}} + \frac{{3 \times 2}}{{2 \times 1}} + 1\\ = 3 + 3 + 1\\ = \underline{\underline 7} \end{array}\)

    \(\begin{array}{c}b)\qquad C_2^4 \times C_2^5 = \frac{{4{\rm{ }}!}}{{2{\rm{ }}!{\rm{ }} \times 2{\rm{ }}!}} \times \frac{{5{\rm{ }}!}}{{3{\rm{ }}!{\rm{ }} \times 2{\rm{ }}!}}\\ = \frac{{4 \times 3}}{{2 \times 1}} \times \frac{{5 \times 4}}{{2 \times 1}}\\ = \underline{\underline {60}} \end{array}\)

    \(\begin{array}{c}c)\qquad C_5^9 \div C_7^9 = \frac{{9{\rm{ }}!}}{{4{\rm{ }}!{\rm{ }} \times 5{\rm{ }}!}} \div \frac{{9{\rm{ }}!}}{{2{\rm{ }}!{\rm{ }} \times 7{\rm{ }}!}}\\ = \frac{{9 \times 8 \times 7 \times 6}}{{4 \times 3 \times 2 \times 1}} \times \frac{{2 \times 1}}{{9 \times 8}}\\ = \underline{\underline {\frac{7}{2}}} \end{array}\)

Solve problems on the combination of distinct objects without repetition

  • Theory
  • Examples
    a) To choose 4 letters from the word INTERVAL, how many combinations are there?

    b) There are 23 members in a soccer team. 11 members are selected among them for
    every match. If 4 of the members must be selected and 2 of the members must not
    be selected, how many combinations of members are there for the next match?
  • Solutions

    a) Number of combinations\(\begin{array}{l} \\= C_4^8\\ = {\underline{\underline {70}} ^{}}\end{array}\)

    b) Number of combinations\(\begin{array}{l} \\= C_7^{17}\\ = {\underline{\underline {19{\rm{ }}448}} ^{}}\end{array}\)