nick1816 wrote:
Players from only two countries \(C_1\) and \(C_2\) participated in a UNO tournament. Each game of the tournament involved 3 players and no two games had the same set of 3 players. The number of games involving only \(C_1\) players was 455 and the number of games involving only \(C_2\) players was 120. If the total number of games played in the tournament was maximum possible, then how many games involved at least one player each from \(C_1\) and \(C_2\)?
A. 1650
B. 1675
C. 1700
D. 1725
E. 1750
So, the game can involve any of the players of any of the two countries..
Since 455 games were played by C1 country, this should give us the number of players of C1..
Let the number of players be n, so \(nC3=455..\frac{.n(n-1)(n-2)}{3!}=455=5*7*13...n(n-1)(n-2)=5*7*13*2*3=13*14*15...n=15\)
Since 120 games were played by C2 country, this should give us the number of players of C2..
Let the number of players be x, so \(xC3=120..\frac{.n(n-1)(n-2)}{3!}=120=4*3*10...n(n-1)(n-2)=3*4*10*2*3=8*9*10...x=10\)
Now there are two ways to have atleast one player of each team..
1) 2 from C1 and 1 from C2 = 15C2*10C1=15*7*10=1050
2) 1 from C1 and 2 from C2 = 15C1*10C2=15*5*9=675
Total = 1050+675=1725
D
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