An executive board must arrange 7 books side-by-side on a display shelf: 3 identical copies of a Finance textbook, 2 identical copies of an Economics textbook, and 2 distinct novels (Novel A and Novel B). How many distinct linear arrangements of these 7 books are possible such that Novel A and Novel B are not placed adjacent to each other?
Answer: 300
Answer
300
To find the number of distinct arrangements where Novel A and Novel B are not adjacent, use complementary counting. First, compute the total distinct arrangements of the 7 books with identical items: 7! / (3! * 2!) = 420. Next, find the number of arrangements where Novel A and Novel B ARE adjacent by bundling them as one unit [AB], which has 2 internal orderings (AB or BA). Arranging the 6 units ([AB], 3 Finance, 2 Economics) gives 6! / (3! * 2!) = 60 ways. Multiplying by 2 internal orderings gives 120 adjacent arrangements. Subtracting these from the total yields 420 - 120 = 300.
Step-by-Step Solution
Key Concept
Counting permutations with indistinguishable items and non-adjacency restrictions using complementary counting
Estimated Time:2m 0s