Shelving five books with two kept apart

Question

You arrange 5 distinct books in a row on a shelf, but two specific volumes are so heavy that they must not be placed next to each other. How many arrangements keep those two books apart?

Accepted Answer

Count by complement: (all arrangements) minus (arrangements where the two are together). All arrangements of 5 distinct books: 5! = 120. Arrangements with the two together (block trick): glue them into one unit, giving 4 units to arrange as 4! = 24, times 2 for their internal order, so 24 	imes 2 = 48. Apart: 120 - 48 = 72. Takeaway. "Not adjacent" is easiest as total minus together: n! - (n-1)!\cdot 2. Here 120 - 48 = 72.

Shelving five books with two kept apart

Your answer

More Probability questions