AMC 8 · 2004 · #2

Grade 3 counting

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

permutations-basicsystematic-enumerationdigit-constraintsplace-value systematic-enumerationcasework ↑ Prerequisites: place-valuesystematic-enumeration

📏 Short solution 💡 2 insights

📘 View easy version →

Problem

Take the four digits of 2004 — namely 2, 0, 0, 4 — and rearrange them. How many different four-digit numbers can you make?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Make a Systematic List

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.

1STEP 1

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.

Order: thousands digit 2 first, then 4

2STEP 2

Put 2 in front; the leftover {0, 0, 4} gives exactly three numbers: 2004, 2040, 2400.

2004, 2040, 2400

3STEP 3

Put 4 in front; the leftover {0, 0, 2} gives three more numbers: 4002, 4020, 4200.

4002, 4020, 4200

4STEP 4

The two lists don't overlap, so add them: 3 + 3 = 6 numbers in all.

3 + 3 = 6 → (B)

Answer

Quick sanity check by a different count. All four digits can be ordered in 4! = 24 ways, but the two 0s are identical, so divide by 2! = 2 to get 12 distinct digit strings. Half of those start with 0 (by symmetry — 0 is just as likely to land in front as any specific digit when all 4 positions are equally available; more precisely, of the 12 strings, exactly 6 start with 0 since after fixing 0 in front the remaining {0,2,4} has 3! = 6 arrangements). So the valid four-digit numbers number 12 - 6 = 6, matching the list and answer (B).

💡Key takeaway

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.

More like this