You say "sort of like the following." Is that one of the cases you are looking at? I don't quite get what you mean by your description of "each set perfectly distributed" unless you just mean "two that start with a, two that start with b, two that start with c, ..."
To begin this problem, though, it is an easy way to think about how you get the 60 permutations. This draws on how permutations relate (similar or different) from combinations. In combinations, order doesn't matter. Thus, we have the three letters, in your case, of "abc". The permutation, however, is not concerned with order. Thus, we have "abc", but we also have "acb", "bac", "bca", "cab", "cba". Thus, each of the 10 combinations expands into a set of 6 permutations on that one combination. Now, if our goal is to get a set of ten that start with certain letters, we'll have to pick from amongst the different expansions of the combinations, since each combination lacks two of the other required letters in our set of 10 permutations we want to define. I don't think it is that difficult, though. If you notice how I rearranged "abc" it is systematic. We can expand every combination in like manner to produce a 6x10 grid. Each of the rows will contain 10 combinations unique from each of the other rows. From there you just need to swap "terms" to meet the requirements you have.