Five cities P, Q, R, S and T are connected by different modes of transport as follows: P and Q are connected by boat as well as rail. Sand R are connected by bus and boat. Q and T are connected by air only. P and R are connected by boat only. T and R are connected by rail and bus. Between which two cities among the pairs of cities given below are there maximum travel options available?

To find the maximum travel options, we need to count the total number of distinct direct and indirect paths (through one intermediate city) available between each pair of cities.

First, list the direct connections and the number of modes for each:

1. P and Q: boat, rail (2 modes)
2. S and R: bus, boat (2 modes)
3. Q and T: air (1 mode)
4. P and R: boat (1 mode)
5. T and R: rail, bus (2 modes)

Now, let's analyze each option:

A) Q and S * Direct connection: 0 options. * Indirect via P and R: Q-P (2 modes) -> P-R (1 mode) -> R-S (2 modes). Total paths = 2 * 1 * 2 = 4 paths. * Indirect via T and R: Q-T (1 mode) -> T-R (2 modes) -> R-S (2 modes). Total paths = 1 * 2 * 2 = 4 paths. * Total for Q and S = 0 + 4 + 4 = 8 options.

B) P and R * Direct connection: P-R (1 mode). Total paths = 1. * Indirect via Q and T: P-Q (2 modes) -> Q-T (1 mode) -> T-R (2 modes). Total paths = 2 * 1 * 2 = 4 paths. * Total for P and R = 1 + 4 = 5 options.

C) P and T * Direct connection: 0 options. * Indirect via Q: P-Q (2 modes) -> Q-T (1 mode). Total paths = 2 * 1 = 2 paths. * Indirect via R: P-R (1 mode) -> R-T (2 modes). Total paths = 1 * 2 = 2 paths. * Total for P and T = 0 + 2 + 2 = 4 options.

D) Q and R * Direct connection: 0 options. * Indirect via P: Q-P (2 modes) -> P-R (1 mode). Total paths = 2 * 1 = 2 paths. * Indirect via T: Q-T (1 mode) -> T-R (2 modes). Total paths = 1 * 2 = 2 paths. * Total for Q and R = 0 + 2 + 2 = 4 options.

Comparing the total options: A) Q and S: 8 options B) P and R: 5 options C) P and T: 4 options D) Q and R: 4 options

Q and S have the maximum travel options (8).

The final answer is A