A chess competition was organized by World Champion Vishwanathan Anand…
2024
A chess competition was organized by World Champion Vishwanathan Anand involving some boys and girls, every student had to play exactly one game with every other student. It was found that in 45 games both the players were girls, and in 190 games both were boys. Then in how many games was one player a boy and the other a girl?
- A.
40
- B.
200
- C.
180
- D.
120
Show answer & explanation
Correct answer: B
In a round-robin tournament, each pair of players meets exactly once, so the number of games among n players equals the combination C(n,2) = n(n-1)/2. When the players split into two disjoint groups (girls and boys), the same-gender games are counted by applying this formula within each group separately, while the mixed games (one from each group) can also be counted directly as the product of the two group sizes, since every member of one group plays every member of the other exactly once.
Let g be the number of girls and b be the number of boys. The girls-only games satisfy C(g,2) = g(g-1)/2 = 45, so g(g-1) = 90; testing consecutive integers, 10 x 9 = 90, so g = 10.
The boys-only games satisfy C(b,2) = b(b-1)/2 = 190, so b(b-1) = 380; testing consecutive integers, 20 x 19 = 380, so b = 20.
Total students = g + b = 10 + 20 = 30, and the total number of games played by all students is C(30,2) = 30 x 29 / 2 = 435.
The games with one boy and one girl equal the total games minus the two same-gender totals: 435 - 45 - 190 = 200.
As an independent check, each of the 10 girls plays each of the 20 boys exactly once, so the mixed games also equal g x b = 10 x 20 = 200, confirming the result by a second method.