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If I have $10$ marbles with different colors and I am choosing $3$, this is a easy problem of just using the combinatoric equation $\dbinom{10}{3} = 120$.

But how would you go about counting the number of combinations if we initially group the marbles on size and we must choose one marble from each group?

So assume we have now $10$ marbles, there are only $3$ sizes, small, medium, and large.

  • Large: $3$ marbles (RED, BLUE, GREEN)
  • Medium: $4$ Marbles (YELLOW, BROWN, BROWN, WHITE)
  • Small: $3$ Marbles (CYAN, GRAY, ORANGE)

How many $3$ marble color combination are there if we had to select one from each group (small, medium, and large)?

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You must pick one small marble, one medium marble, and one large marble. Those are all the choices you have to make, and there are $3, 4,$ and $3$ ways respectively to pick a marble. Therefore the number of 3-marble combinations is $3 \cdot 4 \cdot 3 = 36$.

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