Problem 17 · 2025 Math Kangaroo
Medium
Counting & Probability
complementary-countingcareful-counting
A bee, a mouse, a beetle and a cat want to take a group photo, so they line up next to each other. The cat is not allowed to stand next to the mouse. In how many different ways can the animals line up?
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Answer: B — 12
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Hint 1 of 2
Count all line-ups, then remove the bad ones where the cat and mouse are side by side.
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Hint 2 of 2
Treat the cat-and-mouse pair as a single block to count the forbidden arrangements.
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Approach: total minus the cat-next-to-mouse cases
- Four animals can line up in 4! = 24 ways.
- Glue cat+mouse together: 3 items in 3! = 6 orders, times 2 for the internal order = 12 forbidden line-ups.
- Allowed line-ups = \(24 - 12 = 12\), which is (B).
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