After completing this section, you should be able to:
- Apply the Multiplication Rule for Counting to solve problems.
One of the first bits of mathematical knowledge children learn is how to count objects by pointing to them in turn and saying: “one, two, three, …” That’s a useful skill, but when the number of things that we need to count grows large, that method becomes onerous (or, for very large numbers, impossible for humans to accomplish in a typical human lifespan). So, mathematicians have developed short cuts to counting big numbers. These techniques fall under the mathematical discipline of combinatorics, which is devoted to counting.
Multiplication as a Combinatorial Short Cut
One of the first combinatorial short cuts to counting students learn in school has to do with areas of rectangles. If we have a set of objects to be counted that can be physically arranged into a rectangular shape, then we can use multiplication to do the counting for us. Consider this set of objects (Figure 7.3):
Certainly we can count them by pointing and running through the numbers, but it’s more efficient to group them (Figure 7.4).
If we group the balls by 4s, we see that we have 6 groups (or, we can see this arrangement as 4 groups of 6 balls). Since multiplication is repeated addition (i.e., ), we can use this grouping to quickly see that there are 24 balls.
Let’s generalize this idea a little bit. Let’s say that we’re visiting a bakery that offers customized cupcakes. For the cake, we have three choices: vanilla, chocolate, and strawberry. Each cupcake can be topped with one of four types of frosting: vanilla, chocolate, lemon, and strawberry. How many different cupcake combinations are possible? We can think of laying out all the possibilities in a grid, with cake choices defining the rows and frosting choices defining the columns (Figure 7.5).
Since there are 3 rows (cakes) and 4 columns (frostings), we have possible combinations. This is the reasoning behind the Multiplication Rule for Counting, which is also known as the Fundamental Counting Principle. This rule says that if there are ways to accomplish one task and ways to accomplish a second task, then there are ways to accomplish both tasks. We can tack on additional tasks by multiplying the number of ways to accomplish those tasks to our previous product.
Using the Multiplication Rule for Counting
Every card in a standard deck of cards has two identifying characteristics: a suit (clubs, diamonds, hearts, or spades; these are indicated by these symbols, respectively: , , , ) and a rank (ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king; the letters A, J, Q, and K are used to represent the words). Each possible pair of suit and rank appears exactly once in the deck. How many cards are in the standard deck?
Since there are 4 suits and 13 ranks, the number of cards must be (Figure 7.6).
Using the Multiplication Rule for Counting for 4 Groups
The University Combinatorics Club has 31 members: 8 seniors, 7 juniors, 5 sophomores, and 11 first-years. How many possible 4-person committees can be formed by selecting 1 member from each class?
Since we have 8 choices for the senior, 7 choices for the junior, 5 for the sophomore, and 11 for the first-year, there are different ways to fill out the committee.
Using the Multiplication Rule for Counting for More Groups
The standard license plates for vehicles in a certain state consist of 6 characters: 3 letters followed by 3 digits. There are 26 letters in the alphabet and 10 digits (0 through 9) to choose from. How many license plates can be made using this format?
Since there are 26 different letters and 10 different digits, the total number of possible license plates is .