Raoul's Workout Routine

Solution 1: With no restrictions, there are $5! = 120$ ways for Raoul to do his exercises. Since the exercises are done in an order, for any pair of them, one will always be first. We let $C$ be the number of orders with chin ups first and we let $B$ be the number of orders with bench press first. Since there are 120 total orders, we know that $B + C = 120$ . If we start with an order that has bench press before chin ups, we can swap the order of these two and get an order where chin ups come before bench press. If we have two different orders where bench press is first and we swap the places of bench press and chin ups in both of them, we will get two different orders back. This means that for each order with bench press first, we can find a unique order with chin ups first. We can conclude then, that $B \leq C$ , since for every order with bench press first, we can find a unique order with chin ups first. We can similarly show that $C \leq B$ , and so we must have $C = B$ . Combining this with our equation that $B+C = 120$ we see that $B = C = 60$ .

Solution 2: Out of the 5 exercises, we choose 2 slots for the Chinups and Bench press. There are ${5 \choose 2 } = 10$ ways to do so, since Chinups have to come before the Bench press. For the remaining 3 slots, there are $3 ! = 6$ ways to arrange the other exercises. Hence, by the rule of product, there are $10 \times 6 = 60$ different ways to organize his workout.