Problem 1 · AMC 8 Stretch
Core
Counting & Probability
and-process-multiplyvisual-representation
A man has 3 shirts and 4 ties. In how many ways can he choose a shirt and a tie?
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Answer: 12 outfits
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Hint 1 of 3
He makes the outfit in two steps: first pick a shirt, then pick a tie. The little word 'and' between the two steps is a big clue.
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Hint 2 of 3
When you do one thing AND then another, you multiply the number of choices at each step.
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Hint 3 of 3
Multiply (number of shirts) times (number of ties).
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Approach: AND process — multiply the choices at each step
- Picking an outfit happens in two steps: choose a shirt AND then choose a tie. When you do one thing and then another, multiply.
- There are 3 shirts and 4 ties, so the number of outfits is \(3 \times 4 = 12\).
- You can see it with a tree: draw one branch for each shirt, and let each shirt branch split into 4 ties. Counting the 12 branch-ends gives the 12 outfits.
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