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Having two sets like

$S_1 = \{ A, B, C \}$

$S_2 = \{ X, Y\}$

is fairly simple to understand. You can join every item from $S_1$ with every item from $S_2$ to get the possible combinations. There are $3$ of them in $S_1$ and $2$ of them in $S_2$, so $3$ with X and $3$ with Y. So $3 + 3$ combinations, $3$ times $2 = 6$.

$S_{combinations} = \{ AX, BX, CX, AY, BY, CY \} $

Am I thinking about it the right way thus far?

If so, what happens when you add more sets? Say...

$S_1 = \{ A, B, C \}$

$S_2 = \{ X, Y\}$

$S_3 = \{ M, N, O\}$

The only way I could wrap my head around that is to combine the first two sets:

$S_{1 x 2} = \{ AX, BX, CX, AY, BY, CY \} $

and then combine it with the $S_3$. Is there a better way to think about multiple sets, should I just try to trust the product rule based on its "good behavior"?

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Your way of thinking about this is perfectly fine.

Another approach is the following:

Draw the elements of your three sets in three different parts of the page. Then each choice of an element from each set corresponds to a triangle having one vertex in each part of the page.

Now how many such triangles are there? I think you will be able to see that it is equal to the product of the sizes of the three sets.

Can you see now what the situation looks like for an arbitrary finite number of sets?

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