Problem 21 · AMC 8 Stretch
Core
Counting & Probability
or-process-addand-process-multiply
A state allows plates that are 2 letters followed by 2 digits, OR 2 digits followed by 2 letters. Each digit slot may be 0-9. How many plates are possible?
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Answer: 135,200 plates
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Hint 1 of 3
Two arrangements that don't overlap: letters-first and digits-first. That's an OR process (add).
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Hint 2 of 3
Each arrangement is an AND process. A digit slot here allows 0-9, so 10 choices each.
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Hint 3 of 3
Each format is \(26 \times 26 \times 10 \times 10\). Add the two.
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Approach: OR of two AND processes
- Two non-overlapping formats (OR, add), each an AND process. A digit slot ranges 0-9 (10 choices).
- Letters then digits: \(26 \times 26 \times 10 \times 10 = 67600\). Digits then letters: \(10 \times 10 \times 26 \times 26 = 67600\).
- Add: \(67600 + 67600 = 135200\).
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