AMC 10 · 2020 · #4
Grade 4 number-theoryPick an answer.
The four digits are independent once we pin down their allowed sets — Tool #7 splits the count into four subproblems (one per digit position). Tool #2 lists the allowed values for each position so we don't miss or double-count. Tool #3 eliminates illegal values: odd digits at every position, and 0 at the thousands place. Multiplying the four counts gives the answer in one line.
Fix the units digit
The two rules leave only zero.
Two constraints overlap to leave a single legal digit.
4.OA.B.4Eliminate PossibilitiesCount the leading digit
The leading digit cannot be zero.
A 4-digit number must start with a non-zero digit.
2.NBT.A.1Make A Systematic ListCount the hundreds digit
A middle digit has five choices.
Any even digit works in a non-leading slot.
2.NBT.A.1Make A Systematic ListCount the tens digit
The tens digit is the same.
Tens place follows the same even rule, no leading constraint.
2.NBT.A.1Make A Systematic ListMultiply the counts
Multiply the four slot counts.
Independent choices multiply — like a 4-slot decision tree.
Independent digit choices multiply, like a four-slot decision tree.
▸ Why?
Each slot is filled without regard to the others, so every combination is possible and the counts multiply.
▸ Why?
A number is its digits sitting in fixed places, so choosing the digits is choosing the number.
Match the choice
The result is 100.
Read the matching answer choice.
3.OA.A.3Eliminate PossibilitiesThis AMC 12 problem only needs Grade 4 'multiples of 5 end in 0 or 5' plus place value you already know — units must be 0, leading digit picks from {2, 4, 6, 8}, middle two pick from all five even digits, so 4 × 5 × 5 × 1 = 100.