Problem 7 · AMC 8 Stretch
Stretch
Counting & Probability
and-process-multiplyconsidering-extreme-cases
A key blank has 4 positions where metal can be left alone or cut. At each position you may leave it alone OR cut it at one of 3 different depths. How many different keys can be made if at least one position must be cut?
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Answer: 255 keys
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Hint 1 of 4
Handle each position as its own step in an AND process.
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Hint 2 of 4
At one position you can: leave it alone, or cut at depth 1, or depth 2, or depth 3. How many outcomes is that for one position?
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Hint 3 of 4
Each position has 1 (leave alone) + 3 (depths) = 4 outcomes. There are 4 positions, so multiply 4 four times.
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Approach: AND process per position, then subtract the all-blank key
- Each of the 4 positions is a step. At a position you can leave the metal alone or cut at one of 3 depths, so \(1 + 3 = 4\) outcomes per position.
- Multiply over the 4 positions: \(4 \times 4 \times 4 \times 4 = 4^4 = 256\).
- This counts the key with no cuts at all. Since at least one position must be cut, remove that single key: \(256 - 1 = 255\).
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