AMC 10 · 2017 · #17
Grade 7 arithmeticPick an answer.
AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The phrase "how many" calls for Tool #2 (Make a Systematic List), but listing thousands of numbers by hand is hopeless, so Tool #16 (Change Focus) reframes the count: instead of building numbers digit by digit, choose which digits to use. Once a set of distinct digits is chosen, there is exactly one way to write them strictly increasing and exactly one way strictly decreasing — the order is forced. Tool #7 (Identify Subproblems) splits the work into the increasing case and the decreasing case, and Tool #5 (Look for a Pattern) supplies the count of subsets: a set of n items has 2ⁿ subsets.
Choose the digits, not the number
A monotonous number is fixed by its digit-set and direction, each set ordering just one way — so count digit-sets, not numbers.
Once you know which digits are used, their order is forced, so a digit-set is the same as one number per direction.
7.SP.C.8Count The ComplementCount the increasing numbers
Increasing forbids 0 (it would have to lead), leaving digits 1–9; every nonempty subset gives one number, so 2⁹-1=511.
Each of the 9 usable digits is independently in or out, and only the all-out choice makes no number.
6.EE.A.1Make A Systematic ListCount the decreasing numbers
Decreasing lets 0 ride at the tail, so all digits 0–9 are usable; drop the empty set and the lone {0}, leaving 2¹⁰-2=1022.
Zero can ride along only at the tail, so the decreasing case uses one more digit than the increasing case.
6.EE.A.1Make A Systematic ListCombine without double counting
Adding gives 511+1022=1533, but the 9 single-digit numbers sit in both lists, so subtract them: 1533-9=1524 — choice (B).
Anything counted in both lists must be subtracted once so it ends up counted a single time.
5.OA.A.1Identify SubproblemsPick which digits to use and the order is forced, so just count the digit-sets: 511 going up plus 1022 going down, minus the 9 single digits counted twice, equals 1524 — choice (B).
- Choose the digits, not the number
- Count the increasing numbers
- Count the decreasing numbers
- Combine without double counting