Restrictions on positions

For an arrangement, mark the restricted positions first. Count the choices for those positions before filling the remaining places. Five distinct books include a dictionary. With the dictionary fixed at the left end, the remaining four books give 4!=24 arrangements. If it may occupy either end, the two disjoint cases give 2 x 4!=48. To keep two named people together in a line, treat the pair as one temporary unit, then count its two internal orders. Among five people the count is 4! x 2!=48. For “not together”, subtract the together count from the unrestricted count: 5!-2!4!=72. Every arrangement is in exactly one of these two cases. Check that a case has not been counted twice. All objects here are distinct and used at most once.