Problem 1. A menu offers a choice of 3 salads, 8 main dishes, and 5 desserts. How many different meals consisting of one salad, one main dish, and one dessert are possible?
Problem 2. Wing has different books to arrange on a shelf: 4 blue, 3 green, and 2 red.
a. In how many ways can the books be arranged on a shelf?
b. If books of the same color are to be grouped together, how many arrangements are possible?
c. In how many distinguishable ways can the books be arranged if books of the same color are identical but need not be grouped together?
d. In how many ways can you select 3 books, one of each color, if the order in which the books are selected does not matter?
Problem 3. Some students find it puzzling that 0! = 1, and think that should equal 0. If this were true, what would be the value of (P(4,4) using the permutations formula?
Problem 4. Automobile Manufacturing An automobile manufacturer produces 7 models, each available in 6 different exterior colors, with 4 different upholstery fabrics and 5 interior colors. How many varieties of automobile are available?
Problem 5. Telephone Numbers How many 7-digit telephone numbers are possible if the first digit cannot be zero and
a. only odd digits may be used?
b. the telephone number must be a multiple of 10 (that is, it must end in zero)?
c. the telephone number must be a multiple of 100?
d. the first 3 digits are 481?
e. no repetitions are allowed?
Problem 6. A group of 3 students is to be selected from a group of 12 students to take part in a class in cell biology.
a. In how many ways can this be done?
b. In how many ways can the group who will not take part be chosen?
Problem 7. Worker Grievances A group of 7 workers decides to send a delegation of 2 to their supervisor to discuss their grievances.
a. How many delegations are possible?
b. If it is decided that a particular worker must be in the delegation, how many different delegations are possible?
c. If there are 2 women and 5 men in the group, how many delegations would include at least 1 woman?
Problem 8. Baseball If a baseball coach has 5 good hitters and 4 poor hitters on the bench and chooses 3 players at random, in how many ways can he choose at least 2 good hitters?
Problem 9. Pizza Varieties A television commercial for Little Caesars pizza announced that with the purchase of two pizzas, one could receive free any combination of up to five toppings on each pizza. The commercial shows a young child waiting in line at Little Caesars who calculates that there are 1,048,576 possibilities for the toppings on the two pizzas.*
Verify the child’s calculation. Use the fact that Little Caesars has 11 toppings to choose from. Assume that the order of the two pizzas matters; that is, if the first pizza has combination 1 and the second pizza has combination 2, that is different from combination 2 on the first pizza and combination 1 on the second.