A new airlines company is planning to start operations in a country. The company has identified ten different cities which they plan to connect through their network to start with. The flight duration between any pair of cities will be less than one hour. To start operations, the company has to decide on a daily schedule.

The underlying principle that they are working on is the following:

Any person staying in any of these 10 cities should be able to make a trip to any other city in the morning and should be able to return by the evening of the same day.

**1. If the underlying principle is to be satisfied in such a way that the journey between any two cities can be performed using only direct (non-stop) flights, then the minimum number of direct flights to be scheduled is:**

- 45
- 90
- 180
- 135

**2. Suppose three of the ten cities are to be developed as hubs. A hub is a city which is connected with every other city by direct flights each way, both in the morning as well as in the evening. The only direct flights which will be scheduled are originating and/or terminating in one of the hubs. Then the minimum number of direct flights that need to be scheduled so that the underlying principle of the airline to serve all the ten cities is met without visiting more than one hub during one trip is:**

- 54
- 120
- 96
- 60

Suppose the 10 cities are divided into 4 distinct groups G1, G2, G3, G4 having 3, 3, 2 and 2 cities respectively and that G1 consists of cities named A, B and C. Further, suppose that direct flights are allowed only between two cities satisfying one of the following:

1. Both cities are in G1

2. Between A and any city in G2

3. Between B and any city in G3

4. Between C and any city in G4

**3. Then the minimum number of direct flights that satisfies the underlying principle of the airline is:**

Suppose the 10 cities are divided into 4 distinct groups G1, G2, G3, G4 having 3, 3, 2 and 2 cities respectively and that G1 consists of cities named A, B and C. Further, suppose that direct flights are allowed only between two cities satisfying one of the following:

1. Both cities are in G1

2. Between A and any city in G2

3. Between B and any city in G3

4. Between C and any city in G4

However, due to operational difficulties at A, it was later decided that the only flights that would operate at A would be those to and from B. Cities in G2 would have to be assigned to G3 or to G4.

**4. What would be the maximum reduction in the number of direct flights as compared to the situation before the operational difficulties arose?**

### Answers

- C
- C
- 40
- 4

1. For any pair of cities, say A and B, to satisfy the underlying principle, there must be a morning flight from A to B, an evening flight from B to A and a morning flight from B to A and an evening flight from A to B. Only then can a person from A or B can travel to B or A and return the same day. Hence, there must be four flights between any pair of cities.

Number of ways of selecting two cities from ten cities = (10 × 9)/2 = 45

Hence, the minimum number of flights that must be scheduled = 45 × 4 = 180

2. Let the ten cities be represented by A through J. Among these ten cities, consider A, B and C to be hubs and the other seven cities to be non-hub cities. It is given that any direct flight should originate and/or terminate at a hub.

Consider city D, which not a hub. D should be connected to each of A, B and C. Between D and each of A, B and C, there must be four flights (from the above solution). Hence, from D, there must be 4 × 3 = 12 flights to the three hubs, A, B and C. Similarly, for each of the other six non-hub cities, there must be 12 flights connecting each non-hub city with the three hubs. Hence, a total of 12 × 7 = 84 flights will connect a non-hub city with a hub.

In addition to this, the three hubs must be connected among themselves. Since there must be four flights between any pair of cities, there must be a total of 4 × 3 = 12 flights connecting any pair of hubs.

Hence, the total minimum number of flights that should be scheduled = 84 + 12 = 96.

3. Given that G1 has the cities A, B and C. G2, G3 and G4 have 3, 2 and 2 cities respectively. From the given conditions, we can see that a city in G2 cannot be connected by a direct flight to a city in G3 or G4. Hence, for a person to travel from a city in G2 to a city in G3 or G4, all the cities in G2 must be connected to A and from A, he can travel to B or C to travel to a city G3 or G4 respectively.

Hence, the 3 cities in G2 must be connected to A. Between each pair of cities there must be four flights. Hence, there must be 4 × 3 = 12 flights between cities in G2 and A.

Since there are 2 cities in G3, there must be 2 × 4 = 8 flights between cities in G3 and B.

Since there are 2 cities in G4, there must be 2 × 4 = 8 flights between cities in G4 and C.

Also, the cities in G1, i.e., A, B and C must be connected to each other. Hence, there must be an additional 4 × 3 = 12 flights between these three cities.

Therefore, the total minimum number of direct flights that must be scheduled = 12 + 8 + 8 + 12 = 40

4. It is given that the cities in G2 will be assigned to G3 or G4. However, this, by itself, will not result in any reduction in the number of flights because the cities in G2 will still have to be connected to either B or C.

However, it is also given that there are now no flights between A and C. Hence, the 4 flights that would have been scheduled in the previous case, will now not be scheduled. Hence, the reduction in the number of flights can be a maximum of 4.