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How many ways are there to group 4 people into 2 groups of 2?

Solution: ${4 \choose 2}/2!$

How many ways are there to assign 186 students to 3 rooms, with exactly 62 students in each room?

Solution: ${186 \choose 62}*{124 \choose 62}*{62 \choose 62}$

Why in the second one do we not divide the result by 3! to divide out all the same groups of 3 like we did in the first example?

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    Because presumably the rooms are labeled since they occupy different physical locations and we consider the arrangement where someone is in one room as compared to another to be different. In the groups example, the groups are presumably unlabeled and the experience doesn't change if the same four people were together in the "first group" as when they were in the "second group" together. – 2017-02-17
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    In the first one, you only care about the groups, not their arrangement, so some arrangements are identical. In the second you divide into groups, but the arrangement of the group matters, since they are assigned a room and it matters which room they get. – 2017-02-17
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    If the actual physical location is irrelevant and you care only about who else is in the room with you, then yes you would divide by $3!$. – 2017-02-17

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