Seven different books are to be stood side by side on a shelf. Two of them, an atlas and a dictionary, must NOT be placed next to each other. In how many orders can the seven books be stood?

Count everything and take away the banned orders. All orders of 7 different books: 7! = 5040. Orders in which the atlas and the dictionary ARE neighbours: glue them into one unit, leaving 6 units in 6! = 720 orders, and the two can stand either way round inside the unit, so $720\times 2=1440$. Subtract: 5040 − 1440 = 3600. Trap 1440 answers the opposite condition, 'the two must be together'. Trap 5040 forgets the condition. Trap 2520 = 5040/2 halves the total on the loose idea that 'they are apart half the time'; in fact they are adjacent in 1440 of the 5040 orders, comfortably under half.