Combinatorics practice problems.


Ex.1 A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night for a year?

(1) 5
(2) 6
(3) 7
(4) 8

Sol. (4)
Suppose that the restaurant offers M main courses. Since the choosing of appetizer, main course, and dessert are independent events, there are:

(Appetizers) × (Main courses) × (Desserts) = (2M) × (M) × (3) = 6M²

distinct ways to choose a meal. To cover all possible years, we need to find the smallest integer value of M such that:

6M² ≥ 366, that is, M² ≥ 61, so M ≥ 8.


Ex.2 Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookies on the tray. How many different assortments of six cookies can be selected?

(1) 22  (2) 25  (3) 27  (4) 28

Sol. (4)

Simple application of stars and bars method where n = 6 and k = 3.


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