Problem 22 · AMC 8 Stretch
Stretch
Counting & Probability
Logic & Word Problems
and-process-multiplylogical-reasoning
A state uses plates of 3 letters followed by 3 digits, and every possible plate is already used. To make more plates cheaply, it will add ONE more character slot. Should the new slot be a letter or a digit to create the most new plates? Explain.
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Answer: Add a letter (26 choices)
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Hint 1 of 3
Adding one slot multiplies the number of plates by however many choices that slot has. So you want the slot with the most choices.
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Hint 2 of 3
A letter slot has 26 choices; a digit slot has only 10. Which multiplies your plates more?
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Hint 3 of 3
Compare multiplying by 26 versus multiplying by 10.
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Approach: Adding a slot multiplies by the slot's choice count
- Adding one extra slot multiplies the number of plates by the number of symbols that slot allows. So to make the MOST new plates, choose the slot with the most options.
- A letter offers 26 choices; a digit offers only 10. Since 26 is bigger than 10, adding a LETTER slot makes far more plates.
- The old pool was \(26^3 \times 10^3 = 17{,}576{,}000\) plates. A letter multiplies by 26, giving \(456{,}976{,}000\); a digit would only multiply by 10, giving \(175{,}760{,}000\). So add a letter.
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