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In this chapter, we are trying to develop counting techniques that will be used in the [link] to study probability. One of the most fundamental of such techniques is called the Multiplication Axiom. Before we introduce the multiplication axiom, we first look at some examples.
If a woman has two blouses and three skirts, how many different outfits consisting of a blouse and a skirt can she wear?
Suppose we call the blouses .
We can have the following six outfits.
Alternatively, we can draw a tree diagram:
The tree diagram gives us all six possibilities. The method involves two steps. First the woman chooses a blouse. She has two choices: blouse one or blouse two. If she chooses blouse one, she has three skirts to match it with; skirt one, skirt two, or skirt three. Similarly if she chooses blouse two, she can match it with each of the three skirts, again. The tree diagram helps us visualize these possibilities.
The reader should note that the process involves two steps. For the first step of choosing a blouse, there are two choices, and for each choice of a blouse, there are three choices of choosing a skirt. So altogether there are possibilities.
If, in the above example, we add the shoes to the outfit, we have the following problem.
If a woman has two blouses, three skirts, and two pumps, how many different outfits consisting of a blouse, a skirt, and a pair of pumps can she wear?
Suppose we call the blouses and , the skirts , , and , and the pumps , and .
The following tree diagram results.
We count the number of branches in the tree, and see that there are 12 different possibilities. This time the method involves three steps. First, the woman chooses a blouse. She has two choices: blouse one or blouse two. Now suppose she chooses blouse one. This takes us to step two of the process which consists of choosing a skirt. She has three choices for a skirt, and let us suppose she chooses skirt two. Now that she has chosen a blouse and a skirt, we have moved to the third step of choosing a pair of pumps. Since she has two pairs of pumps, she has two choices for the last step. Let us suppose she chooses pumps two. She has chosen the outfit consisting of blouse one, skirt two, and pumps two, or . By looking at the different branches on the tree, one can easily see the other possibilities.
The important thing to observe here, again, is that this is a three step process. There are two choices for the first step of choosing a blouse. For each choice of a blouse, there are three choices of choosing a skirt, and for each combination of a blouse and a skirt, there are two choices of selecting a pair of pumps. All in all, we have different possibilities.
The tree diagrams help us visualize the different possibilities, but they are not practical when the possibilities are numerous. Besides, we are mostly interested in finding the number of elements in the set and not the actual possibilities. But once the problem is envisioned, we can solve it without a tree diagram. The two examples we just solved may have given us a clue to do just that.
Let us now try to solve [link] without a tree diagram. Recall that the problem involved three steps: choosing a blouse, choosing a skirt, and choosing a pair of pumps. The number of ways of choosing each are listed below.
| The Number of ways of choosing a blouse | The number of ways of choosing a skirt | The number of ways of choosing pumps |
| 2 | 3 | 2 |
By multiplying these three numbers we get 12, which is what we got when we did the problem using a tree diagram.
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