Competition · AMC preparation · step 4 of 4
AMC 8 · 2004 · #2
Grade 3 countingPick an answer.
AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
There are only four digits to shuffle, and the leading digit can only be 2 or 4. The whole answer space is tiny, so the cleanest approach is Tool #2 (Make a Systematic List): pick an ordering rule, write every valid number in that order, and count. No formulas needed — the list itself is the proof.
Decide the ordering rule
Fix an order first: the thousands digit can only be 2 or 4 (never 0), so list the numbers starting with 2, then those starting with 4.
A clear rule before listing prevents missed cases and duplicates — the heart of Tool #2.
3.OA.A.3Make A Systematic ListList the numbers starting with 2
Put 2 in front; the leftover {0, 0, 4} gives exactly three numbers: 2004, 2040, 2400.
Fixing the leading digit reduces the problem to placing one nonzero digit among two identical zeros — easy to enumerate.
3.OA.A.3Make A Systematic ListList the numbers starting with 4
Put 4 in front; the leftover {0, 0, 2} gives three more numbers: 4002, 4020, 4200.
The structure mirrors the previous case — same shape, different leading digit.
3.OA.A.3Make A Systematic ListCombine and count
The two lists don't overlap, so add them: 3 + 3 = 6 numbers in all.
The Addition Principle: when two case lists do not overlap, the total is the sum of their sizes.
The count of all valid four-digit numbers equals the count of those that start with 2 plus the count of those that start with 4.
▸ Why?
The numbers starting with 2 and the numbers starting with 4 together cover every valid number, and no number belongs to both groups, so the two group counts add back to the whole count.
▸ Why?
A valid four-digit number cannot begin with 0, and the only digits on hand are 2, 0, 0, 4, so the first digit must be 2 or 4 — placing every valid number in exactly one of the two groups.
▸ Why?
If the leading place held 0 it would count no thousands, so the string would really be a shorter number rather than a four-digit one — the first digit is never 0.
▸ Why?
When a collection is split into non-overlapping groups that leave nothing out, the sizes of the groups add back to the size of the whole collection.
When the answer is a small number, just list them all in a sensible order. Here only 2 or 4 can lead, and each leading digit gives 3 numbers — total 6.
- Decide the ordering rule
- List the numbers starting with 2
- List the numbers starting with 4
- Combine and count
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