Counting Principle, Permutations & Combinations

Name:

1.Recall or listen to the song “The Twelve Days of Christmas”. How many gifts have accumulated after the 12th day?

2.Dominoes are rectangular tiles used to play a game. Each tile is divided into two squares with a number of dots in each square, as in the figure. The standard set of dominoes has from 0 to 6 dots in each square. A deluxe set of dominoes has from 0 to 9 dots in each square.

a.) How many different standard dominoes are possible?

b.) How many different deluxe dominoes are possible?

c.) How many dominoes are possible in a set that has 0 to n dots in each square?

3.Most radio stations in the United States have 3 or 4 call letters.

a.) Historically, radio stations west of the Mississippi river had call letters starting with K. How many different sets of call letters like this are possible?

b.) Today, radio station letters begin with K or W. How many different call letters like this are possible?

4.Recall the Boston Market article. The counting problem discussed in the article is to determine the number of different 3-side-dish choices that can be made from 16 dishes.

a.) Are repetitions allowed in this situation? Does order matter? Are the 3-side-dish choices examples of permutations, combinations, or neither?

b.) The correct answer given by the math teacher and his students is 816. Explain how to determine this answer.

5.Examine the following TV commercial that aired nationwide:

a.) Suppose you order just one pizza and you must choose exactly five different toppings from 11 choices. How many different pizzas are possible?

b.) Suppose you order just one pizza and you must choose exactly three different toppings from 11 choices. How many different pizzas are possible?

c.) Suppose you order just one pizza and you can choose from 0 to 5 different toppings. How many different pizzas are possible?

d.) In the TV commercial, does the 4-year-old boy have the correct answer? If so, explain how to compute his answer. If not, explain why it is incorrect and determine the correct answer.

e.) Jeremy reasoned as follows:

There are 1024 possibilities for one pizza. Since 2 pizzas are ordered, that makes (1024)2 possibilities for a two-pizza order. But order doesn’t matter for the two pizzas, so divide by 2. Thus, the correct answer is 524,288.

Explain the error in Jeremy’s reasoning.