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The Rule of Product

*Math 381 Blog # 1*

How many shirt and pants combinations can I wear if I have 7 shirts and 5 pairs of pants? This and other questions can be answered by the basic concepts of combinatorics. We will be using something called *The Rule of Product*. According to Arthur T. Benjamin, “The rule of product states that if an action can be performed by making A choices followed by B choices, then it can be performed A*B ways.”

Let’s look at the shirt and pants problem first. Again, we have 7 shirts (A shirts):

and 5 pairs of pants (B pairs of pants)

Forget fashion for a minute. How many possible outfits could you have? Well, think about this. If we pick the pink shirt, then we have 5 choices of pants to wear with that shirt. Since there are 7 shirts and 5 pairs of pants, we must use the rule of products (A*B) to give us \(7\cdot 5=35\) different outfit combinations.

Now let’s think of another problem. Say I have 12 hats

How many shirt, pants, and hat combinations are there?

Let’s list off what we know:

There are 7 shirts There are 5 pairs of pants There are 12 hats

We know that shirt and pants combinations can be stated as \(7\cdot 5=35\)

So in order to include the hat combinations, we would state this as \(7\cdot 5\cdot 12=455\) different outfit combinations.

Now it’s time for the suggested

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