is the "birthday paradox"-like one there to catch out those that have been training for these questions, spot a birthday question, immediately calculate for 10*9 pairs rather than just 9?
It certainly seems to be some sort of trick question, but it couldn't be to catch the mistake you mention: The bet is a losing proposition even if you win $1 for each person sharing any other's birthday, not just your own. It remains a losing proposition even if you bring the number of people up above 23 (the birthday paradox number) and reverse the 1:2 payoff and assume the bettor has no special information.
Perhaps the kind of solution they're looking for is something along the lines of "Only if I knew that seven of the other people's birthdays matched mine." Or maybe this is simply one of those "impossible" questions designed to test your creativity.
