🇺🇸 AMC 8 ⇄ switch contest
1999 AMC 8

Problem 15

Problem 15 · 1999 AMC 8 Hard
Counting & Probability multiplication-principleoptimization

Bicycle license plates in Flatville each contain three letters. The first is chosen from the set {C, H, L, P, R}, the second from {A, I, O}, and the third from {D, M, N, T}. When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set, or one letter may be added to one set and one to another. What is the largest possible number of additional license plates that can be made by adding two letters?

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Answer: D — 40 more plates.
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Hint 1 of 2
Each plate is one choice from each set, so the count is the PRODUCT of the three set sizes: 5 × 3 × 4 = 60. Adding a letter bumps one factor up by 1.
Still stuck? Show hint 2 →
Hint 2 of 2
Adding to a set multiplies the whole product by that factor's growth — going 3→4 multiplies by 4/3, going 5→6 only by 6/5. So push your new letters into the smallest sets to get the biggest multiplier.
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Approach: count = product of set sizes; feed the smallest factors
  1. Plates = (size 1)×(size 2)×(size 3) = 5 × 3 × 4 = 60. Two new letters add 2 total across the three factors.
  2. A letter helps most where the factor is smallest, because it multiplies by the larger ratio. The size-3 set is smallest: putting both there gives 5 × 5 × 4 = 100; splitting one into the 3-set and one into the 4-set gives 5 × 4 × 5 = 100 as well.
  3. Either way that's 100 plates, an extra 100 − 60 = 40. The principle: to grow a product of fixed total "+1 bumps," feed your increments to the smallest factors — they give the steepest percentage jump.
Another way — just test every placement (brute force, when in doubt):
  1. Six ways to place the two letters; compute each product. Both into set 1: 7×3×4 = 84. Both into set 2: 5×5×4 = 100. Both into set 3: 5×3×6 = 90.
  2. Split 1&2: 6×4×4 = 96. Split 1&3: 6×3×5 = 90. Split 2&3: 5×4×5 = 100.
  3. The max is 100, so the gain is 100 − 60 = 40. With only six cases, exhaustively checking is perfectly safe and confirms the smallest-factor logic.
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