Question

In a class tournament where the participants were to play one game with another, two class players fell ill, having played 3 games each. If the total number of games played is 84, the number of participants at the beginning was..........

Answer 1

15.
Suppose the two players did not play at all so that the remaining (n-2) players played ${}^{n-2}{C}_2$
matches. Since these two players played 3 matches each, hence the total number of matches is ${}^{n-2}{C}_2$ + 3 + 3 = 84 (given)
$\frac{(n-2)(n-3)}{12}=78$
or $n^2-5n+6=156$ or $n^2$ - 5n - 150 = 0
or (n-15) (n+10) = 0 $\therefore$ n = 15 $(n\ne -10)$