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suppose I have 3 different color of bags, yellow, blue, red, each contain 2 marbles that are the same color as the bag.

I know the number of combination of picking 2 different color marble out of 3 bags is just 3 pick 2 = 3. But what if the marbles are labeled with number, like this:

Yellow(1), Yellow(2)  - bag 1
Blue(1), Blue(2)      - bag 2
Red(1), Red(2)        - bag 3

How many combination are there for picking 2 different color with the label in mind? That means: Yellow(1) + Blue(1), Yellow(1) + Blue(2) Yellow(2) + Blue(1), Yellow(2) + Blue(2)... There are 12 ways but I am not sure what is the formula I can use.

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Suppose you pick any one of the marble then you have $2$ ways to select the different marble

Example: If you pick $\color{red}{\text{Red(1)}}$ then you can pick $\color{yellow}{\text{Yellow(2)}}$ or $\color{blue}{\text{Blue(2)}}$.

And you can pick any of $6$ marbles. Means $6$ marbles each having $2$ choice therefore $$\text{Total number of ways}=6\times2\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=12 \text{ ways}$$

EDIT: (If the marbles are not numbered)

As if only color remains different then if you select any color marble then you have $2$ ways to pick a different marble of different color again you have $6$ marbles then $$\text{Total number of ways}=6\times2\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=12 \text{ ways}$$

(If the marbles are numbered)

If marbles are numbered and only different colored marbles are to be pick. Now, you have $4$ choices with each marble $$\text{Total number of ways}=6\times4\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=24\text{ ways}$$ As the cases such as $\{Yellow(1)+Blue(1)\},\{Blue(1)+Yellow(1)\}$ are counted so we need to divide by $2$ $$\text{Total number of ways}=\frac{24}{2}$$

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  • $\begingroup$ @user1701840 Discuss here chat.stackexchange.com/rooms/56810/we-are-friends $\endgroup$ – Harsh Kumar Apr 18 '17 at 1:08
  • $\begingroup$ what if I have 1 RED, 2 YELLOW, 1 BLACK, 2 BLUE? Then I should think like this: If I pick 1 RED, then I have 3 ways of select different marble. And I can pick any of the 6 marbles, so I am looking at 6 * 3 = 18 ways, but it is not correct. $\endgroup$ – user1701840 Apr 18 '17 at 1:12
  • $\begingroup$ not sure if I understand that, I can pick Red1 and Black 1, that is the third way $\endgroup$ – user1701840 Apr 18 '17 at 1:16
  • $\begingroup$ But according to question you can't select the same number. $\endgroup$ – Harsh Kumar Apr 18 '17 at 1:24
  • $\begingroup$ sorry I edited my question to clarify $\endgroup$ – user1701840 Apr 18 '17 at 1:27

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