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 exam problem:

Given the same numbers of shirts and pants as in the example above, how many outfit combinations are possible if I just want to either use the cowboy hat or the fireman hat?