A reliable method
Identify how many choices exist at each stage. Multiply for independent sequential choices; use combinations when the order of a chosen group does not matter.
The common trap
An arrangement of three people into roles differs from choosing three people for an unlabelled team. Double-counting can hide inside familiar formulas.
Make the connection
Try a tiny case that can be listed by hand. For a sum of an arithmetic sequence, pair first and last terms to check the averaging formula.
Worked example: choose the requested quantity
A code consists of one of 3 letters followed by one of 4 digits. Repetition restrictions do not apply. How many codes are possible? For each possible letter, every digit is allowed. Use the product rule for the two stages. The requested value is 12.
Contrast and diagnose an error
How many different pairs can be chosen from 4 people if order does not matter? There are n(n−1) ordered choices of two different people. Divide by 2 because each unlabelled pair was counted in both orders. The requested value is 6. A tempting answer is 4: Choose one person instead of a pair.
Transfer practice and check
What is the sum of the first 3 positive odd integers? Try it before reading on. Check: 9. The sequence begins at 1 and ends at twice the number of terms minus 1. Multiply the term count by the average of the first and last terms.
A code consists of one of 3 letters followed by one of 4 digits. Repetition restrictions do not apply. How many codes are possible?
- For each possible letter, every digit is allowed.
- Use the product rule for the two stages.
- The requested value is 12.
Compare the other answer choices
Add choices instead of multiplying stages.
Use the letter count at both stages.
Count each two-stage code twice.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content