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How many ways can you distribute 5 marbles in 3 identical ba

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How many ways can you distribute 5 marbles in 3 identical baskets such that each basket gets at least 1 marble.

I am trying to solve this problems using as many approaches:

The correct answer is:
we can pick distribute the marbles in two such ways:
3-1-1 or 2-2-1.
For the 3-1-1. We choose 3 marbles for the 1st basket. (5C3). 1 for the 2nd basket (2C1) and from the remaining marble (1C1).
We then divide by 2!. I know it has something to do with the the fact we are putting 1 marble into the 2 baskets but what is the intuition?? How do you if you have to divide by n!?? This is what I am not getting. After we have (5C3)(2C1)(1C1) does the problem become... " How many ways can we rearrange the digits 311?" Ans: 3!/(2!1!)

If I choose 3 people out of 5 for a soccer game its 5*4*3/3!. I divide by 3! Because order does not matter. How does this translate to the above problem?

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