AMC 10 · 2017 · #11
Grade 7 countingPick an answer.
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
The choice is a set of digits, not a number.
Once you know which digits are used, their order is forced, so a digit-set is the same as one number per direction.
Once you know which digits are used, their order is forced, so a digit set names exactly one number.
▸ Why?
Each set can be arranged in the required order in exactly one way, so sets and numbers pair up perfectly.
▸ Why?
Each digit is independently in or out of the set, so counting sets is counting yes-or-no choices.
Count the increasing numbers
An increasing number cannot use zero.
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
A decreasing number may put zero last.
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
Removing the shared single digits gives 1524.
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