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9 different books are to be arranged on a bookshelf. 4 of these books were written by Shakespeare, 2 by Dickens, and 3 by Conrad. How many possible permutations are there if the books by Conrad must be separated from one another?

 Dec 19, 2019
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The number of ways to order Shakespeare's and Dickens' book is 6!. There will be 7 spaces (in front of all the books, in between the first and second book, in between the second and third book...all the way until at the end of all the books) that Conrad's books can go with them since they can't be next to each other. The number of ways of doing this is binom{7}{3}⋅3! since once they are in their spots they can be ordered 3! ways.

Thus, the number of ways is 6!⋅binom{7}{3}⋅3!=151200.

 Dec 20, 2019

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