X red balls indistinguishable and Y green balls indistinguishable Let's say we have 10 (X) red balls and 3 (Y) green balls. All the balls are indistinguishable.
Firstly, does "indistinguishable" mean that the order of the balls doesn't matter since they do not have a number?
Secondly :


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*How many ways can we place these 13 balls in a line?

*How many ways can we place these 13 balls in a line, if only their color matters?

*How many ways can we choose 3 red balls and 2 green balls?


For the question number one, I think it is $$ 13! $$
For two other questions, I have no idea. I do not grasp the reasoning, the logic behind.
What if we have K colors?

Yes indistinguishable means it doesn't matter if you swap two indistinguishable balls, since you can't distinguish the two results (but I assume only in question 2. you mean balls of the same color are indistuingishable and for the rest the balls are distinguishable)
Assuming you mean distinguishable balls (else the second question formulation wouldn't make sense) some hints:


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*Your answer is correct: For the first position you have 13 choices of balls (since they are distinguishable). For the second position 12 and so on. This yields your solution.

*Hint: When only the color matters think of putting 13 balls in one line. Now choose which three you paint yellow. So it's choose 3 out of 13 (to be yellow) possibilities (I hope you know binomial coefficients).

*Hint: How many ways can you choose 3 balls out of 10, how many ways 2 out of 3? Now multiply these numbers (why?).

