Problem:
The cards in a deck are numbered . Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked. The two persons with lower numbered cards form a team, and the two persons with higher numbered cards form another team. Let be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards and , and Dylan picks the other of these two cards. The minimum value of for which can be written as , where and are relatively prime positive integers. Find .
Solution:
Alex and Dylan are on the same team if Blair and Corey picked cards numbered and with either or from the cards from the deck excluding the cards numbered and .
Thus
Because , it follows that
and thus
Hence . Because is an integer, it follows that or ; that is, or . Thus the minimum possible value of is equal to
and the requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions