Problem 11 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiply
A carrier has 3 letters and 5 mailboxes. In how many ways can the letters be placed if (a) each letter must go in a different mailbox; (b) any number of letters may share a mailbox?
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Answer: (a) 60; (b) 125
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Hint 1 of 4
Place the letters one at a time — placing each letter is a step in an AND process.
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Hint 2 of 4
In part (b), each letter independently has all 5 mailboxes to choose from. In part (a), a box already used by an earlier letter is no longer available.
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Hint 3 of 4
(b): \(5 \times 5 \times 5\) (three letters, 5 choices each). (a): 5 for the first letter, then 4, then 3, as boxes get used up.
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Approach: AND process, placing letters one at a time
- Place the 3 letters one after another — a 3-step AND process.
- (a) Different mailboxes: the first letter has 5 choices, the second has 4 (one box is taken), the third has 3, so \(5 \times 4 \times 3 = 60\).
- (b) No restriction: each letter independently may go in any of the 5 boxes, so \(5 \times 5 \times 5 = 125\).
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