Problem 30 · 2025 Math Kangaroo
Stretch
Counting & Probability
caseworkcareful-counting
There are 12 children at a party, including 3 pairs of twins. How many different ways are there to distribute six blue hats and six red hats to the children, so that each pair of twins wears hats of the same colour?
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Answer: C — 92
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Hint 1 of 2
Treat each twin pair as a single unit that takes two hats of one colour; six singletons take one hat each.
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Hint 2 of 2
Sum over how many pairs are blue, then choose colours for the singletons to balance to 6 and 6.
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Approach: casework on the number of blue twin-pairs
- If k of the 3 pairs wear blue, they use 2k blue hats, leaving 6−2k blue hats for the six singletons.
- Ways = Σ C(3,k)·C(6, 6−2k) for k=0..3 = 1 + 45 + 45 + 1.
- Total = 92 ways.
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