Count Distinct Ordered Roles

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: From 5 distinct candidates choose a chair and a different recorder. How many assignments are possible? Candidate answers: (A) 19 (B) 20 (C) 21 (D) 40.
• Independent check inputs: From 6 distinct candidates choose a chair and a different recorder. How many assignments are possible? Candidate answers: (A) 29 (B) 30 (C) 31 (D) 60.

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. Choosing people for different roles makes order matter. After filling the first role, that person cannot fill the other stated distinct role, so the second choice count decreases by one.

Here is the method on the supplied teaching case. There are 5 choices for chair, then 4 for recorder. Ordered roles distinguish assignments, so multiply to get 20.

Chair-and-recorder assignments differ when the people exchange roles. Do not divide by two in this ordered problem; the no-repeat requirement reduces the second-stage options rather than the first-stage ones.

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.

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Take it further

Changed-input task: From 12 distinct candidates choose a chair and a different recorder. How many assignments are possible? Candidate answers: (A) 131 (B) 132 (C) 133 (D) 264. Show the mathematical reason, not only the selected answer. Expected result: 132. There are 12 choices for chair, then 11 for recorder. Ordered roles distinguish assignments, so multiply to get 132.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.