Problem 16 · 2018 AMC 8
Medium
Counting & Probability
careful-countingcasework
Professor Chang has nine different language books lined up on a bookshelf: two Arabic, three German, and four Spanish. How many ways are there to arrange the nine books on the shelf keeping the Arabic books together and keeping the Spanish books together?
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Answer: C — 5760 ways.
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Hint 1 of 2
"Must stay together" is a gift: tape each must-stick group into a single fat book. Now you're arranging far fewer objects — but the German books aren't taped, so they each stay separate.
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Hint 2 of 2
The technique is the block / glue method: arrange the blocks-and-loose-items as one layer, then multiply by the internal arrangements inside each block (the books in a block can still shuffle among themselves).
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Approach: block-then-internal
- Glue the 2 Arabic books into one block and the 4 Spanish books into another. The shelf now holds 5 movable objects: [Arabic block], [Spanish block], and the 3 separate German books — arranged in 5! = 120 ways.
- Don't forget the books can rearrange inside their blocks: the Arabic block has 2! = 2 internal orders, the Spanish block has 4! = 24.
- Total: 120 × 2 × 24 = 5760.
- You'll see it again: every "these items must be adjacent" problem is glue-the-group-into-one, arrange, then × the internal orderings — and items that aren't required to be together simply stay as their own objects.
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