Xavier Aptitude Test 2011 Paper

(d) No. of teams in the bottom group = 10No. of matches played among the bottom group teams = $^{10}$$C_2$= 45$\therefore$ No. of points that bottom group getsplaying amongst themselves = 45 (1 pt per match).It is given in the question that the bottom teams get 45 points from their matches against the top group , i.e.45 out 10 n points where n is the no. of teams in the top group. No. of points that the top group gets from the matches played among themselves = ${}^n C_2$No. of points that the top group gets against the bottom group = 10 n – 45${}^n C_2 = 10 n - 45$$n(n+1)=20n-90$$n^{2} - 21 n + 90 = 0$$(n-6)(n-15)=0$If $n=6,$ the top group would get ${}^n C_2 + 10 n - 45$If $n=6,$ the top group would get ${}^n C_2 + 10 n - 45$$= {}^n C_2 + 10(6) - 45 = 30$ points or an average of 5 points per team the bottom group would get $\frac{(45+45)}{10}$ or an avg of $9 .$ This is not possible as points of the top group should be higher than that of the bottom group. $\therefore n=15 .$ $\therefore$ Total no. of teams $=25$