Read the information given below and answer the question that follows.

Ten teams participated in a hockey tournament, in which every team played every other team once. After the tournament was over, it was found that the ten teams could be divided into 2 groups A and B, such that every team in A won against every team in B and every team in A won the same number of matches. If every team in A won 6 matches, how many teams were there in B?

#### Options

2

3

4

5

#### Solution

Let the number of teams in team B be b, then the number of teams in A is (10 - b).

Corresponding to every match played between the teams in A, there is a win for A. Corresponding to every match played between an A-team and a B-team, there is a win for A. Therefore the total number of A Vs A. matches and A Vs B matches is equal to the number of wins for A.

Total number of matches played = Total number of matches won by teams in A + the number of matches played by the teams in B among themselves.

⇒ `""^10C_2 = 6(10 - b) + ""^bC_2`

⇒ 45 = 60 - 6b + `(b(b - 1))/(2)`

⇒ b^{2} – 13b + 30 = 0

⇒ b = 3 or 10.

Since b < 10, b = 3.**3 **is the correct option.