Problem: In a tennis tournament, n women and 2n men play, and each player plays exactly one match with every other player. If there are no ties and the ratio of the number of matches won by women to the number of matches won by men is 7/5, then n equals
Answer Choices:
A. 2
B. 4
C. 6
D. 7
E. none of these
Solution:
Let k denote the number of matches in which women defeated men, and let W and M denote the total number of inatches won by women and by men, respectively. The total number of matches. the number of matches between a man and a woman, the number of matches between two men and the number of matches between two women are
23n(3n−1),2n2,22n(2n−1), and 2n(n−1)
respectively. Then. since k≤2n2, one has
2k+n(n−1)3n(3n−1)=2W2(M+−WW)+1=7M63n2−21n=24k+12n2−12n817n2−3n=k≤2n217n2−3n≤16n2n≤3.
Substituting n=1,2,3 yields k=14/8,62/8,144/8. Since k must be an integer, n=3.