Problem:
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people so that exactly one person receives the type of meal ordered by that person.
Solution:
Choose one person to receive the correct meal; this can be done in ways. Then select people to receive the other two meals of that type. If these meals are both given to people who originally ordered the same type of meal, there are ways of choosing these people, and the remaining meals are then completely determined. If these two other meals go to people who ordered different types of meals, all remaining meals are determined except for the other diners who should have gotten the type of meal that was served correctly. There are ways to choose the recipients of the first two of these meals, and then ways to distribute the last meals. Thus there are ways to distribute the meals.
The problems on this page are the property of the MAA's American Mathematics Competitions