Question

P, Q, R and S are four towns. One can travel between P and Q along 3 direct paths, between Q and S along 4 direct paths, and between P and R along 4 direct paths. There is no direct path between P and S, while there are few direct paths between Q and R, and between R and S. One can travel from P to S either via Q, or via R, or via Q followed by R, respectively, in exactly 62 possible ways. One can also travel from Q to R either directly, or via P, or via S, in exactly 27 possible ways. Then, the number of direct paths between Q and R is

Answer 1

Correct Answer: 7

Let the number of direct paths between Q and R be x, and the number of direct paths between R and S be y.

From the given information, we can form the following equations: 1. Number of ways to travel from P to S via Q = 3 × 4 = 12 2. Number of ways to travel from P to S via Q and R = 3 × y = 4y 3. Number of ways to travel from P to S via R = 4 y = 4y The total number of ways to travel from P to S is 62, so: 12 + 4y + 3xy = 62

Similarly, for the paths from Q to R: 1. Number of ways to travel from Q to R directly = x 2. Number of ways to travel from Q to R via P = 3 × 4 = 12 3. Number of ways to travel from Q to R via S = 4 × y = 4y

The total number of ways to travel from Q to R is 27, so: x + 12 + 4y = 27

Solving these two equations, we get x = 3.

Therefore, the number of direct paths between Q and R is 3.