5p3 Permutation - Perfectly distributed Sets

Okay, I'm looking for a mathematical solution that may or may not exist. So I am asking smarter minds than mine. (I'm not a pure math guy, so my terminology may be all off.)

I'm starting with 5 items taken 3 at a time, and I'm looking for a way to match up the permutations in six sets.

5C3 gives me 10 possible combinations (abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde.)

5P3 gives me 60 possible permutations which I put into an excel spreadsheet.

So, is there a method to break these 60 permutations into 6 unique sets of the 10 combinations, with each set perfectly distributed. (2 apiece starting with ABCDE, 2 apiece with the middle letter ABCDE and 2 apiece with the last letter ABCDE).

Sort of like the following: (abe, acd, bac, bed, cae, cdb, dba, dec eda, ecb) except my middle letters don't pan out and its only one of the six possible sets.

Any directions for a solution? I am totally open to brute force ideas using spreadsheets or an awesome statistical abacus. My current solution is a box of 60 notecard permutations hand-sorted on a table. Man, I should have paid better attention in statistics.