Since no one shakes his own hand or his spouse's, the most handshakes is 6. Since there are seven different answers, then the given answers are 0 through 6. If Mr A shook 6 hands, Mrs A had to be the one who shook no hands. Similarly, if Mr B shook 5 hands, then Mrs B could only have shaken 1 hand; and Mr and Mrs C shook 4 and 2 hands respectively; and Mr and Mrs D would have shaken 3 hands each. Since only one person could have given an answer of 3, then Fred had to be Mr D, meaning that Ethel was Mrs D. This means that Ethel shook 3 hands.
Today's brain teaser courtesy of Braingle.com.