I'd say you first choose three couples from the five, then one person from each chosen couple:
The way you are doing is ok, just remember you have got some duplicates, since you were doing them by order (permutation instead of combination). For example, you could get A C E, and you might also get C A E, and you'd count them as two outcomes while in fact it is the same combination of people in different orders. To get to the correct answer from your approach, you would divide your answer by the total possible ways to put three people in order.
Keep on asking, and it will be given you;
keep on seeking, and you will find;
keep on knocking, and it will be opened to you.