It might be easiest to find out the number of patterns first, then figure out the possible arrangements of those patterns. For example, you could take a pattern like 123231, assuming that lower numbers appear first. You can multiply the number of these patterns by 6 (3!) to account for the actual ways you could match the numbers 1, 2, 3 to the professions E, D, and T, and then finally multiply by a further 8 (23) to account for the arrangements where the people in the same profession are swapped.
Following these rules, the patterns must all start with:
12????
Clearly 2 can't be next. If it's 1, you can work out that there's actually only one possible resulting arrangement. On the other hand if it's 3, then there's a few possibilities, but it's still fairly easy to enumerate them. Then you could just multiply the total number of arrangements by 6*8 = 48 due to the reasoning earlier and you're done.