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Grade 11 · lesson 2 of 4

Remove Reversed Duplicate Committee Selections

An unordered two-person committee does not distinguish first and second selection.

About 25 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: From 5 distinct candidates choose an unordered two-person committee. How many committees are possible? Candidate answers: (A) 10 (B) 11 (C) 20 (D) 9.
  • Independent check inputs: From 6 distinct candidates choose an unordered two-person committee. How many committees are possible? Candidate answers: (A) 14 (B) 15 (C) 16 (D) 30.

Learn the idea

Counting methods distinguish ordered roles from unordered groups. A probability model assigns chances to possible outcomes, and expected value or binomial counting combines those probabilities while preserving what a single trial can actually produce. An unordered two-person committee does not distinguish first and second selection. Ordered counting includes each pair twice, so dividing by two recovers the distinct committees.

Here is the method on the supplied teaching case. Ordered selections total 5(4), but each pair is counted twice in reverse order. Divide by 2: 10 committees.

Only divide by the number of equivalent orderings appropriate to the stated group size. Different named roles would make reversed choices genuinely different; a committee without roles is the unordered case here.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: From 5 distinct candidates choose an unordered two-person committee. How many committees are possible? Candidate answers: (A) 10 (B) 11 (C) 20 (D) 9. Solution: Ordered selections total 5(4), but each pair is counted twice in reverse order. Divide by 2: 10 committees. The final answer is 10. Only divide by the number of equivalent orderings appropriate to the stated group size. Different named roles would make reversed choices genuinely different; a committee without roles is the unordered case here.

Your turn

Check your reasoning

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Helpful tips
  • An unordered two-person committee does not distinguish first and second selection.
  • Only divide by the number of equivalent orderings appropriate to the stated group size.
From 6 distinct candidates choose an unordered two-person committee. How many committees are possible?

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See the answer and explanation

15

Ordered selections total 6(5), but each pair is counted twice in reverse order. Divide by 2: 15 committees.

Take it further

Changed-input task: From 12 distinct candidates choose an unordered two-person committee. How many committees are possible? Candidate answers: (A) 132 (B) 65 (C) 66 (D) 67. Show the mathematical reason, not only the selected answer. Expected result: 66. Ordered selections total 12(11), but each pair is counted twice in reverse order. Divide by 2: 66 committees.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. From 7 distinct candidates choose an unordered two-person committee. How many committees are possible?
Need a hint?
  • An unordered two-person committee does not distinguish first and second selection.
  • Only divide by the number of equivalent orderings appropriate to the stated group size.
2. From 8 distinct candidates choose an unordered two-person committee. How many committees are possible?
Need a hint?
  • An unordered two-person committee does not distinguish first and second selection.
  • Only divide by the number of equivalent orderings appropriate to the stated group size.
3. From 9 distinct candidates choose an unordered two-person committee. How many committees are possible?
Need a hint?
  • An unordered two-person committee does not distinguish first and second selection.
  • Only divide by the number of equivalent orderings appropriate to the stated group size.

Your choices are temporary. Feedback appears after you check all the answers.

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