KA-0072Systematic listing
3 points, difficulty 1 of 3Level 3–4about 45sHow many different two-digit numbers can be written using only the digits 1, 2 and 3, if a digit may be used twice?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
List them in order, smallest first.
2The strategy
Fix the first digit, then run the second digit through all three options. Repeat for each first digit. Listing in that order guarantees nothing is skipped or repeated.
3The full solution
11, 12, 13, 21, 22, 23, 31, 32, 33 — nine numbers, which is 3 choices times 3 choices.
Solution— no shortcut on this one
List by first digit. Starting with 1: 11, 12, 13. Starting with 2: 21, 22, 23. Starting with 3: 31, 32, 33. That is nine numbers. It is also 3 choices for the first digit times 3 for the second, because repeats are allowed and the first choice does not use anything up. The transferable idea: read whether repeats are allowed before you decide what to multiply by.
Why this is on the test: Whether repeats are allowed changes 3 x 3 into 3 x 2, and students often decide it from habit rather than from the sentence.