CAT 2000 — QA Question 49
Answer the following question based on the information given below.
Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.
The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.
There are three cities A, B and C, each of these cities is connected with the other two cities by at least one direct road. If a traveller wants to go from one city (origin) to another city (destination), she can do so either by traversing a road connecting the two cities directly, or by traversing two roads, the first connecting the origin to the third city and the second connecting the third city to the destination. In all there are 33 routes from A to B (including those via C). Similarly there are 23 routes from B to C (including those via A). How many roads are there from A to C directly?
Answer & solution
6
- B
3
- C
5
- D
10
Let there be,
x roads connecting A and B directly,
y roads connecting B and C directly and
z roads connecting C and A directly
∴ Total number of routes connecting A and B: x + yz = 33 …(i)
∴ Total number of routes connecting B and C: y + xz = 23 …(ii)
Subtracting (ii) from (i)
(x − y) + z(y − x) = 10
−1(y − x) + z(y − x) = 10
(y − x)(z − 1) = 10
(y − x)(z − 1) = 5 × 2 …(iii)
From the options, the possible values for z are 3 and 6.
Consider z = 3
∴ y – x = 5 …(iv)
From equations (i), (ii) and (iv) we get the values as x = 4.5 and y = 9.5 which is not possible
Consider z = 6
∴ y – x = 2 …(v)
Solving (i), (ii) and (v), we get y = 5, x = 3
Thus, there are 6 direct roads between A and C.
Hence, option (a).