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?
Explanation:
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).
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