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A shelf is to hold, in a single row, 4 different novels and 3 different biographies, a total of 7 books. Two specific novels form a matched set and must be placed next to each other, and no two biographies may be adjacent to each other. In how many distinguishable orders can the 7 books be arranged?
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Correct answer
C
Glue the two matched novels into a single block. The novels then form 3 units: that block plus the other 2 single novels. Arrange the 3 units in 3! = 6 ways and order the 2 books inside the block in 2! = 2 ways, giving 6 × 2 = 12 novel arrangements. Those 3 units, once placed in a row, create 4 gaps: one before the first unit, one between each pair of adjacent units, and one after the last unit.
To keep the 3 distinct biographies non-adjacent to each other, drop them into 3 of those 4 gaps, one per gap, in an ordered way, since the 3 biographies are different books: P(4,3) = 4 × 3 × 2 = 24. Multiply the stages: 12 × 24 = 288. So the answer is C.
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