# How to interpret the product rule with multiple sets?

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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