Questions: Systematic Listing

Question 1

How many two-digit numbers can be formed using the digits {1, 2, 3} if digits can be repeated?

• A. 3
• B. 6
• C. 9
• D. 12

Answer: (C)

Systematically: fix the first digit and list all second digits. First digit 1: 11, 12, 13. First digit 2: 21, 22, 23. First digit 3: 31, 32, 33. That is 9 numbers total. Alternatively, 3 choices for the first digit × 3 choices for the second digit = 9. The systematic listing confirms the multiplication principle.

Question 2

When listing most possible outcomes systematically, the order in which you list them affects whether the list is complete.

• T. True
• F. False

Answer: False

Any systematic order (alphabetical, numerical, by fixed variable) will produce the same complete list — the order of listing does not change what is on the list. What matters is that the system covers every possibility without gaps or duplicates. Different orderings are equally valid as long as they are exhaustive.

Question 3

List all subsets of {a, b, c} systematically.

Model answer: Systematically by size: Size 0: {}. Size 1: {a}, {b}, {c}. Size 2: {a,b}, {a,c}, {b,c}. Size 3: {a,b,c}. Total: 8 subsets.

Organizing by size guarantees completeness. Within each size, listing in alphabetical order ensures no subset is missed or duplicated. The total count 2³ = 8 confirms the listing is complete. This systematic approach — fix one organizing principle (size), then systematically vary within that (alphabetical order) — scales to larger sets.