Solution

Among the five married couples no one shook more than eight hands. Therefore, if nine people each shake a different number of hands, the numbers must be 0, 1, 2, ..., and 8. The person who shook 8 hands has to be married to the person who shook 0 hands (otherwise that person could have shaken only seven hands.) Similarly, the person who shook seven hands is bound to be married to the person who shook 1 hand. So that the married couples shook hands in pairs 8/0, 7/1, 6/2, 5/3. The only person left who shook hands with 4 is my wife.

Reference