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Product Rule

Let's say a person chooses a red, green, white, or blue shirt and black, brown, or grey pants. How many different outfits can they create? 

We could just list every single variation like this:

Red shirt and black pants

Red shirt and brown pants

Red shirt and grey pants

Green shirt and black pants


To make things easier to read, we could also create a tree diagram like this:

Each branch of the tree diagram represents a different outfit we can create, so there are 12 outfits in total.

These techniques do work and is nice to use to visualize what the question is asking, but when we start to see larger numbers or more decisions to make, this method becomes tedious. So as a shortcut, you may notice a relationship between the numbers in our question and answer. We can choose between 4 different shirts and 3 different pants and we end up with 12 different outfits. By observation we see that \( 4 \times 3 = 12 \). This is called the Product Rule.

We use the product rule when we are counting the number of events and each of our choices can occur at the same time. In this case, we are choosing a shirt and a pair of pants to wear at the same time.

Usually, a good indicator for questions like these is the word "and". If one does one action AND another action that can be done at the same time, this is a signal to multiply the number of ways to do these actions together. 

Sum Rule

What if a person can select a top from 3 shirts or 4 sweaters? Assuming a person can't wear a shirt and sweater at the same time, how many variations are there? 

Since we are choosing one top out of 3 shirts and 4 sweaters, we can see that we are choosing out of 7 items in total. Then there are 7 variations. 

In counting, this is the Sum Rule since we are taking the sum of all our options. 

If one action can be done in \(m\) ways and another action can be done in \(n\) ways, if both actions cannot be done at the same time, then there are \(m + n\) possible outcomes.  

In our shirt/sweater example, we had 3 shirts and 4 sweaters, and we could not pick both a shirt and a sweater at the same time. 

A good indicator for questions like this is the word "or". If we can choose from one set of objects OR another, we usually use the sum rule. 

Another example: 

Let's say you are playing a game of chance with a friend. You shuffle a deck of cards and you win if you choose either an ace or the 3 of diamonds. How many winning outcomes are there? 

In a standard deck of cards there are 4 aces and 13 of diamonds. Before thinking to just add 4 and 13, you must remember that the ace of diamonds falls under both of these categories so by adding 4 and 13, you'd be double-counting that card. In this case there are 16 possibilities as

\( 4 + 13 - 1 = 16 \)

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