AMC 10 · 2003 · #10

Grade 6 counting
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
fundamental-counting-principleexponentsratio-proportion identify-subproblems ↑ Prerequisites: fundamental-counting-principle
📏 Medium solution 💡 1 insight
Problem
An old license plate is one letter followed by four digits; a new license plate is three letters followed by three digits. There are 26 possible letters and 10 possible digits, and both may repeat. By how many times has the number of possible license plates increased — that is, what is the new count divided by the old count?

Pick an answer.

(A)
$\frac{26}{10}$
(B)
$\frac{26^2}{10^2}$
(C)
$\frac{26^2}{10}$
(D)
$\frac{26^3}{10^3}$
(E)
$\frac{26^3}{10^2}$

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

A plate is built slot by slot, and each slot is chosen independently, so tool #7 (Identify Subproblems) says: count the choices for each slot and multiply them to get the total number of plates. Do this once for the old scheme and once for the new. Then tool #16 (Change Focus) reframes "increased by how many times" as a single division — new total over old total — rather than a subtraction. Because the answer choices are all ratios of powers of 26 and 10, tool #3 (Eliminate Possibilities) lets the simplified exponents point straight at the matching choice without any large multiplication.

1STEP 1

Count the old plates

Five slots — 26 choices for the letter, 10 for each digit. Multiplying gives 26 · 10⁴ old plates.

26 · 10 · 10 · 10 · 10 = 26 · 10⁴
2STEP 2

Count the new plates

Now three letter slots (26 each) and three digit slots (10 each). Multiplying gives 26³ · 10³ new plates.

26 · 26 · 26 · 10 · 10 · 10 = 26³ · 10³
3STEP 3

Turn "how many times" into a division

"By how many times" asks for new ÷ old, not the difference — so form the single fraction of new over old.

new/old = (26³ · 10³)/(26 · 10⁴)
4STEP 4

Simplify the ratio

Cancel the shared factors: 26³⁻¹ = 26² stays on top, one spare 10 stays below. The ratio is 26²/10 — choice (C).

(26³ · 10³)/(26 · 10⁴) = 26³⁻¹ · 10³⁻⁴ = 26²/10 → (C)
Answer
26²/10
Going from one letter to three letters multiplies the count by 26², and going from four digits to three digits divides it by 10. So the plate count should grow by exactly 26² and shrink by 10, giving 26²/10 — matching (C). The result is bigger than 1 (about 67.6), which makes sense: gaining two extra letter slots ( × 676) far outweighs losing one digit slot ( ÷ 10), so the new scheme really does allow more plates.
💡Key takeaway

Count each plate by multiplying the choices in every slot, then divide the new count by the old to see how many times bigger it got.

  • Count the old plates
  • Count the new plates
  • Turn "how many times" into a division
  • Simplify the ratio

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