Possible webpage title: Understanding Combinations with Identical Objects

[Polymath89]

I have two basic questions about combinations. If there are e.g. 10 objects of which 3 are identical and you want to pick a group of 6 out of those 10, how many groups could you get in this case? I know how basic combinations work, but what if there are identical objects involved?

And my second question refers to these problems:

"In how many ways can you distribute 5 marbles in 3 identical baskets such that each basket has at least 1 marble?" and a variation of that with "3 distinct baskets".

Well for each question there are the possibilities that either one basket gets 3 and the other two 1 or two baskets get 2 and the remaining one 1. Now if the baskets are identical the solution for the 3-1-1 outcome is:

5C3*2C1*1C1/2!

and for the problem in which the baskets are distinct:

3C1*5C3*2C1*1C1

I understand the intuition behind the solution for the problem with the distinct baskets, but I don't understand why you have to divide by 2! on the problem with identical baskets, because I thought that the order does not matter, and 5C3*2C1*1C1 already reflects that the order does not matter, doesn't it?

I'd appreciate a short answer. Thanks in advance.

 

[chiro]

Hey Polymath89.

The easiest way to think about this is to consider the complement where at least one basket doesn't have one marble.

This is given by either (5,0,0) (4,1,0) or (3,2,0) combinations. The first can happen 3C1 times and the other two can happen 3C2 times.

In total we get 3C1 + 2*3C2 = 3 + 12 = 15 different ways of not getting at least 1 in every slot.

I'll think about the number of possibilities in a sec.