Problem:
A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is m/n, where m and n are relatively prime positive integers, find m+n.
Solution:
In order for Terry and Mary to get the same color combination, they must select all red candies or all blue candies, or they must each select one of each color. The probability of getting all red candies is (218)(210)(28)=20⋅19⋅18⋅1710⋅9⋅8⋅7. The probability of getting all blue candies is the same. The probability that they each select one of each color is (220)(218)102⋅92=20⋅19⋅18⋅17102⋅92⋅4. Thus the probability of getting the same combination is
2⋅20⋅19⋅18⋅1710⋅9⋅8⋅7+20⋅19⋅18⋅17102⋅92⋅4=20⋅19⋅18⋅1710⋅9⋅8⋅(14+45)=19⋅172⋅59=323118
and m+n=441.
The problems on this page are the property of the MAA's American Mathematics Competitions