(Q). 30 players from India, Canada & Russia participatedd in world chess tournament. Number of players from india is 3 more than the number of players from canada. In tournament players from same country are not allowed to play against each other & no two player plays against each other more than once. Then find
(i). The maximum number of matches that was played in the tournament
(ii). If number of matches is maximum then what is the ratio of number of players from India & Russia ?
Solution:-
Let number of players from Canada = x
∴ Number of players from India = x + 3
∴ Number of players from Russia = 30 – (x + x + 3)
= 27 – 2x
now from the given condition number of matches are as follows:-
⟹Number of matches between players of India & Canada is = ˣC₁ × ⁽ˣ⁺³⁾C₁ = x.(x+3)
⟹Number of matches between players of Canada & Russia is = ˣC₁ × ⁽²⁷⁻²ˣ⁾C₁ = x.(27 – 2x)
⟹ Number of matches between players of India & Russia is = ⁽ˣ⁺³⁾C₁ × ⁽²⁷⁻²ˣ⁾C₁ = (x+3).(27-2x)
So total number of matches
= x.(x+3) + x.(27-2x) + (x+3).(27-2x)
= x² + 3x + 27x – 2x² + 27x + 81 – 2x² – 6x
= -3x² + 51x + 81
So total number of matches is ( -3x² + 51x + 81) & it will attain a maximum value at
x = =
= 8.5
but x must be integers
So possible value of x is 8 or 9
⦿ When x = 8 then total number of matches
= -3(8)² + 51×8 + 81
= 297
⦿ When x = 9 then total number of matches
= -3(9)² + 51×9 + 81
= 297
Hence maximum number of matches = 297 when x=8 or 9 Answer
(ii).
⦿ when x=8 then players from India & Russia is 11, 11. Hence required ration = 11 ∶ 11 = 1 ∶ 1
⦿ when x=9 then players from India & Russia is 12 & 9. Hence required ratio = 12 ∶ 9 = 4 ∶ 3