Problem 20 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiplyor-process-add
Find the number of plates for each rule: (a) 2 letters followed by a 2-digit number (10-99); (b) 2 letters then a 2-digit number, OR a 2-digit number then 2 letters; (c) 3 characters, each of which can be any letter or any digit.
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Answer: (a) 60840; (b) 121680; (c) 46656
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Hint 1 of 3
A 2-digit number means 10-99, which is 90 values. A letter has 26 choices, a digit 10 choices, and a 'letter or digit' slot has 36 choices.
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Hint 2 of 3
(a) is one AND process. (b) is two formats that don't overlap (OR of two ANDs). (c) lets each of 3 slots be any of 36 symbols.
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Hint 3 of 3
(a): \(26 \times 26 \times 90\). (b): add the two arrangements (each equal to part (a)). (c): \(36 \times 36 \times 36\).
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Approach: AND, OR-of-ANDs, and the 36-symbol slot
- A '2-digit number' means 10-99, which is 90 values. A letter slot has 26 choices, a digit slot 10, and a 'letter or digit' slot \(26 + 10 = 36\).
- (a) 2 letters then a 2-digit number (AND): \(26 \times 26 \times 90 = 60840\).
- (b) That format OR the reverse (two non-overlapping formats; each equals part (a)): \(60840 + 60840 = 121680\).
- (c) 3 characters, each a letter or digit (AND): \(36 \times 36 \times 36 = 46656\).
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