Problem 18 · AMC 8 Stretch
Core
Counting & Probability
or-process-addand-process-multiply
A state makes regular plates two ways. Old plates: 2 letters then a 2-digit number from 10 to 99. New plates: 2 letters then a 3-digit number from 100 to 999. How many regular plates are possible in all?
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Answer: 669,240 plates
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Hint 1 of 4
There are two separate plate formats (old and new). Adding the two counts is an OR process.
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Hint 2 of 4
Each format is built by an AND process. A 2-digit number 10-99 has 90 values; a 3-digit number 100-999 has 900 values.
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Hint 3 of 4
Old: \(26 \times 26 \times 90\) (two letters, then the 2-digit number). New: \(26 \times 26 \times 900\). Add the two.
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Approach: OR of two AND processes
- Two separate formats means an OR process (add), and each format is an AND process (multiply).
- Old plates: 2 letters then a number 10-99 (that's 90 values): \(26 \times 26 \times 90 = 676 \times 90 = 60840\).
- New plates: 2 letters then a number 100-999 (that's 900 values): \(26 \times 26 \times 900 = 676 \times 900 = 608400\).
- Add the two formats: \(60840 + 608400 = 669240\).
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