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**Plato** Actually this is an over count by a factor of (3!).

These are known as unordered partitions.

If the question were “how many ways to choose a red team, blue team, and green team?” then the above answer would be correct. That is because the partitions are ordered by color.

It is curious to note that the correct answer is equal to $\displaystyle {17 \choose 5} \, {11 \choose 5}$.

That is, take the first person listed and then choose five others to be with that person.

Then take the next person on the list not chosen and choose five more to go with that person.

Having done that, the remaining six people constitute the third group.