Voodo,
I can see that you are frustrated and concerned, but believe me that we will not stop until you get this, ok?
That said, I'm going to try to use your dice analogy and extend it to match the sibling problem, ok?
Let's say that we have 7 dice, which have already been rolled. Each die has a serial number etched into it, so that we can tell one die apart from another.
We have no idea what the serial numbers are on the dice (let's assume that the etching is microscopic), but we are told that upon being rolled, the following numbers are shown:
1. Two of the dice show the number 1
2. Two of the dice show the number 2
3. Three of the dice show the number 3
The question we are posed is the following: how many different ways can we select two dice from the group of seven, such that each die we select shows a different face value?
Do you see how the fact that each die has its own serial number has absolutely no bearing on the question?
Similarly, we do not need to know the names of the people in the group of 7...we simply need to know that the sibling relationships exist.
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Now, you can of course enumerate every possibility of names to relationships, but the answer will work out to be the same...kind of like working with a non-reduced fraction.
Example of how this could work with dice...
Quesion: What is the probability that two six-sided dice have the same face value?
Method 1: We realize that it doesn't matter what the first die is, as long as the second die matches the first die.
Therefore, the probabilty of both dice having the same face value is the same as the probability of the second die being the same as the first die. Since the first die can only be one value at a time, that probability is 1/6.
Method 2: We simply map out all possible values of the first die and then all possible values for the second die, count the total matches, and then divide by the grand total of posibilities:
Die 1 is 1
Die 2 can be 1,2,3,4,5, or 6 --> 1 match (1) out of 6 possibilities
Die 1 is 2
Die 2 can be 1,2,3,4,5, or 6 --> 1 match (2) out of 6 possibilities
Die 1 is 3
Die 2 can be 1,2,3,4,5, or 6 --> 1 match (3) out of 6 possibilities
Die 1 is 4
Die 2 can be 1,2,3,4,5, or 6 --> 1 match (4) out of 6 possibilities
Die 1 is 5
Die 2 can be 1,2,3,4,5, or 6 --> 1 match (5) out of 6 possibilities
Die 1 is 6
Die 2 can be 1,2,3,4,5, or 6 --> 1 match (6) out of 6 possibilities
Total matches = 6
Total possibilities = 36
6/36 = 1/6
I encourage you to assign letters to the sibling problem and work it out the way you describe. You will see the symmetry for yourself I am sure.
