Competition · AMC preparation · step 4 of 4
AMC 8 · 2015 · #10
Grade 5 countingPick an answer.
AMC 8 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
A 4-digit number is just four slots to fill — thousands, hundreds, tens, units. Tool #7 (Identify Subproblems) splits the count into four independent choices, one per slot, so the multiplication principle gives the total. To convince ourselves the multiplication is right, Tool #9 (Solve an Easier Related Problem) starts with a 2-digit version we can list by hand, then extends the pattern up to 4 digits.
Try the two-digit warm-up
Warm up on the 2-digit case: 9 tens-digit choices × 9 units-digit choices = 81, confirming the choices-times-choices logic scales up.
Solving the easier 2-digit case first shows that "choices per slot, then multiply" really works.
3.OA.A.1Solve An Easier Related ProblemCount the thousands digit
Slot 1 (thousands): 0 is banned as the leading digit, so there are 9 choices, not 10.
Treating one digit position as its own mini-problem with its own constraint is the Tool #7 subproblems move.
4.OA.A.3Identify SubproblemsCount the hundreds digit
Slot 2 (hundreds): any digit except the thousands one (0 is fine here), so 10 - 1 = 9 choices.
Each slot's count depends on how many digits are still unused; subtract used digits from 10.
4.OA.A.3Identify SubproblemsCount the tens digit
Slot 3 (tens): must differ from the two digits already used, so 10 - 2 = 8 choices.
Same pattern: 10 digits minus those already used.
4.OA.A.3Identify SubproblemsCount the units digit
Slot 4 (units): must differ from the three digits already used, so 10 - 3 = 7 choices.
Last slot has the fewest options because the most digits are already taken.
4.OA.A.3Identify SubproblemsMultiply the four counts
Multiplication principle: multiply the four slot counts, 9 × 9 × 8 × 7 = 4536 → (B).
Each combination of slot choices gives a different number, and every valid number arises this way exactly once.
The count of four-digit numbers with four distinct digits equals the product of the per-slot choice counts, 9 × 9 × 8 × 7.
▸ Why?
Each valid number is built by making four choices in order — a digit for the thousands slot, then hundreds, then tens, then units — and each choice's option count depends only on how many digits are already used, not on which earlier digits were picked, so the choices are independent. Independent choices made in sequence combine into the total by multiplying their per-slot counts.
▸ Why?
That product counts every distinct-digit number once and only once, because each such number matches exactly one allowed sequence of slot-picks and each allowed sequence matches exactly one number.
Big counting problems get easy when you split them into one small choice per slot, then multiply — a Grade 5 expression-evaluation move.
- Try the two-digit warm-up
- Count the thousands digit
- Count the hundreds digit
- Count the tens digit
- Count the units digit
- Multiply the four counts
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