Using the Multiplication Property in Sets

When you want to count up how many things are in a set, you have quite a few options. When the set contains too many elements to count accurately, you look for some sort of pattern or rule to help out. Here, you practice the multiplication property.

If you can do task one in m₁ ways, task two in m₂ ways, task three in m₃ ways, and so on, then you can perform all the tasks in a total of m₁ · m₂ · m_{3 . . .} ways.

Sample questions

1. How many ways can you fly from San Francisco to New York City, stopping in Denver, Chicago, and Buffalo, if the website offers four ways to fly from San Francisco to Denver, six ways to fly from Denver to Chicago, two ways to fly from Chicago to Buffalo, and three ways to fly from Buffalo to New York City?

   144. Multiply 4 x 6 x 2 x 3 = 144. This method doesn’t tell you what all the routes are; it just tells you how many are possible so you know when you’ve listed all of them. (Better get to work on that.)

2. How many ways can you write a password if the first symbol has to be a digit from 1 to 9; the second, third, and fourth symbols have to be letters of the English alphabet; and the last symbol has to be from the set {!, @, #, $, %, ^, &, *, +}?

   1,423,656. You multiply 9 x 26 x 26 x 26 x 9 = 1,423,656. This system allows a lot of passwords, but most institutions make you use eight or more characters, which makes the number of possibilities even greater.

Practice questions

1. If you have to take one class in each subject, how many different course loads can you create if you have a choice of four math classes, three history classes, eight English classes, and five science classes?

2. How many different ice-cream sundaes can you create if you have a choice of five ice-cream flavors, three sauces, and five sprinkled toppings if you choose one of each type?

3. How many different automobiles can you order if you have a choice of six colors, four interiors, two trim options, three warranties, and two types of seats?

4. How many different dinners can you order if you have a choice of 12 appetizers, 8 entrees, 5 potatoes, 6 desserts, and a choice of soup or salad?

Following are answers to the practice questions:

1. The answer is 480.

   Multiply: 4 x 3 x 8 x 5 = 480.

2. The answer is 75.

   Multiply: 5 x 3 x 5 = 75.

3. The answer is 288.

   Multiply: 6 x 4 x 2 x 3 x 2 = 288.

4. The answer is 5,760.

   Multiply: 12 x 8 x 5 x 6 x 2 = 5,760. Don’t forget that soup or salad is two choices for that selection.