Once one accepts that the concept of an empty string is quite reasonable, the next question concerns the answer to the exercise in question. Now consider: are all characters contained in the empty string equal to 1? - The answer is yes, of course. Hence the empty string has to be included in the counting of all strings of length consisting entirely of 1s.
Having said that, the interpretation under which we say that strings consisting only of 1s are those that do not contain any 0s would give a natural case for inclusion of the empty string.