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'm a little bit unsure how to approach this problem of permutations when you add categories:

Each object can contain 3 attributes out of the 10 possible available.

Because order doesn't matter, there would be 10 x 9 x 8 = 720 possible choices. If duplications of attributes don't matter (eg. you can have 3 of the same attribute), then there will be 10 x 10 x 10 possible choices. This part is easy.

But what I don't understand, is what happens if this object can randomly come from any of 30 countries? And what if this object can be of any 5 sizes? (These are not attributes mentioned before)

How many possibilities are there?

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If there are multiple independent choices (the attributes, the country of origin, the sizes), you just multiply the number of possibilities.

Because order doesn't matter, there would be 10 x 9 x 8 = 720 possible choices.

That's not right, actually. That would be the number if order did matter. If order doesn't matter, the number is $\binom{10}{3}=120$.

So if there are $10$ attributes, $3$ are randomly chosen, no repeats, and there can be 30 countries of origin and 5 sizes, then there are

$\displaystyle\binom{10}{3}*30*5=18000$ possibilities.

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