CAT 2000 Question 154

Question 154

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?

Solution

The possible roads are:

A -> B Let this be x
A -> C Let this be y
B -> C Let this be z

From the information given, x + yz = 33 -> (1)
z + xy = 23 -> (2)

From the options, if y = 10, x =2 and z = 3 from (2), but it doesn't satisfy (1)
If y = 5, x = 4 and z = 3 from (2) but they don't satisfy (1)
A possible set of numbers for (x,y,z) are (3,6,5)
Number of roads from A -> C = 6



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