This is a very common problem. I'm copying it from something I typed before, so don't worry if the are transformed into X.
(set of permutations) because the random variables are iid.
Considering the permutation that inverts 1 and k for some k, we obtain that has the same distribution as .
So , the follow the same distribution.
Now consider the function :
and apply it to the vectors .
It follows that are identically distributed for all k.
In particular, it follows that .