In the book "Knowing the Odds - An Introduction to Probability" by John B. Walsh, pp.12-13 he states the following:
first of all, Symmetry Principle says,
Symmetry Principle. If two events are indistinguishable except for the way the outcomes are labeled, they are equally likely.
Now, let's see the first example,
[There are] five identical marbles from one to five, put them in a sack, mix them up, close your eyes, and choose one. the numbers have no influence on the draw since we cannot see them. Then what is the probability that marble one is drawn first? What is the probability that marble five is drawn last?
the answer for these questions are of course 1/5; since all five marbles have equally likely chance (or more clearler, all five marbles by symmetric definition, they have identical chance to choose)
Now for the next example, a pair of dice. he states the following,
[we can] write the outcomes as 36 equally likely ordered pairs. So, for instance, the point 12 can only be made by (6,6), so it has probability 1/36, while the point 11 can be made with (5,6) or (6,5), so it has probability 2/36.
But, in fact, a pair of dice are usually of the same color, and essentially indistinguishable. This led Gottfried Leibniz [...] to write that points 11 and 12 were equally likely. This was because each point can be made in only one way: the 12 by two sixes, the 11 by a six and a five.
This contradicts the previous calculation
he refers to "five identical marbles" example
Rather than laugh at Leibniz, let us restate what he said. In effect, he thought that the only 21 different combinations are equally likely, not the 36 different ordered pairs.
What Leibniz thought when he said there are 21 different combinations are equally likely, not the 36 different ordered pairs ..
and the author continue to define the problem,
Why are the 21 combinations not equally likely? (Hint: Is it possible to relabel the dice to change any given combination into any other? If it is, the symmetry principle would tell us Leibnitz was right, and we would have a genuine problem!)
so the 21 combinations do not satisfy the Symmetry principle, I tried to count these combinations but I don't know how ..
Thanks in advance ..
To count the combinations, ask yourself how many combinations have 6 as the highest number: There are 6 of these. (61, 62, 63, 64, 65, 66)
Then how many combinations have 5 as the highest number: 5 tf those.
4 as highest? 4 of those.
...
Lastly, add up all those combinations: 6+5+4+3+2+1 = 21.
As to how to prove that no relabeling could change any given combination into any other, that is more subtle, and to follow the book all you need to know is that it could be proven.