Shelving books so no two novels touch

Question

You shelve 5 distinct math books and 3 distinct novels in a row so that no two novels are next to each other. How many arrangements are possible?

Accepted Answer

Use the "gap" method. First arrange the 5 math books in a row: 5! = 120 ways. Lined up, they create 6 gaps to slot novels into (one before each book, one between each pair, and one after the last): \,M\,\,M\,\,M\,\,M\,\,M\,\ To keep novels apart, put at most one novel per gap. Choose 3 of the 6 gaps for the novels, \binom{6}{3} = 20, and arrange the 3 distinct novels in those chosen gaps, 3! = 6. Multiply everything: 5! 	imes \binom{6}{3} 	imes 3! = 120 	imes 20 	imes 6 = 14400. Takeaway. For "no two of a group adjacent," place the others first, then drop the group into separate gaps…

Shelving books so no two novels touch

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