Problem 22 · 2020 Math Kangaroo
Stretch
Counting & Probability
Logic & Word Problems
caseworkcareful-counting
Twelve colored cubes are lined up side by side: three blue, two yellow, three red and four green, but not in that order. There is a red cube at one end and a yellow one at the other end. The red cubes are all together, and the green cubes are all together. The tenth cube from the left is blue. In how many ways can the cubes be lined up?
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Answer: D — 9
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Hint 1 of 2
The red block of 3 sits at the red end, the green block of 4 stays together, and position 10 is blue.
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Hint 2 of 2
Place the two big blocks and the end yellow first, then count where the loose blues and the other yellow can go.
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Approach: place the blocks, then count the rest
- Red (3 together) occupies the red end and yellow occupies the far end; green stays as a block of 4.
- With position 10 forced blue, the green block and the remaining blues and yellow have only a few valid placements.
- Counting all consistent arrangements gives 9 ways.
- The answer is 9, choice D.
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