Trouble differentiating questions with repeated arrangements

Risecurve

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Can somebody please clarify the difference between these two questions? I know how to do them but I don't fully understand why you must divide the # of ways the groups can be moved around in the first question but not the second.

1. In how many ways could a class of 18 students divide into groups of 3 students each? (Answer is 190590400)

2. How many ways could 15 different books be divided equally among 3 people? (Answer is 756756)

For question 2, I arrived at 756756 by doing (15C5) x (10C5) x (5C5). Then I proceeded to divide the product by 3! because I thought I had to cancel out the ways the 3 groups repeated in arrangements. What am I doing wrong?
 
You actually do end up dividing to cancel out the non-unique solutions in both problems, although you don't divide by the same number. I believe the answer to why this happens will become apparent if you work through the problem. Take the first one, for instance:

You have 18 students and want to arrange them into 6 groups of 3. The first step is to form just one group. How many ways can you choose 3 students from a pool of 18? Given that you have six groups to be assigned, call that first group Group A. But what if instead you'd called it Group B? It would produce a technically different, but non-unique solution. Can you see why that is? Now, what must be done to account for these non-unique solutions? After that, there's 15 students left. How many ways can you choose 3 students from a pool of 15? Given that you have five groups remaining, you again have some non-unique solutions to account for. How can you do that? Repeat this same process until you run out of students. At the end of the process, you will have taken six steps to cancel out all the non-unique solutions. Collectively, could all those cancellations be done in one calculation? What is that calculation? How does that apply to the similar second problem?
 
Can somebody please clarify the difference between these two questions? I know how to do them but I don't fully understand why you must divide the # of ways the groups can be moved around in the first question but not the second.
1. In how many ways could a class of 18 students divide into groups of 3 students each? (Answer is 190590400)
2. How many ways could 15 different books be divided equally among 3 people? (Answer is 756756)
This maybe just repeating what has been said.
#1 is known as divisions into un-ordered partitions: \(\displaystyle \dfrac{18!}{(3!)^6(6!)}\)

#2 is known as divisions into ordered partitions: \(\displaystyle \dfrac{15!}{(5!)^3}\)
 
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