Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then $$D_{i_1i_2...i_n}f = D_{i{\sigma (1)}i{\sigma (2)}...i{\sigma (n)}}f$$ Basically, prove that we can reorder mixed partials however we like, as long as every partial is continuous.
I get that this is basically the Clairaut theorem extended, but I'm not sure how to extend it appropriately. One thought was just to observe that you can get to any order by a sequence of 'flipping' two adjacent partials -- i.e if we have 123 we can make it 321 through 123 -> 132 -> 312 -> 321. This would involve only 2 at once so, I think, would be just an application of the Clairaut. But I'm not sure how to formalize this.
Your idea is good since adjacent transpositions generate all permutations. So it suffices to show that Clauraut's theorem holds for adjacent transpositions.