Question

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:

• A. 45
• B. 90
• C. 180
• D. 135
• A. 54
• B. 120
• C. 96
• D. 60

Answer 1

The Correct Option is C

Minimum Flights Required Between 10 Cities

Step 1: Flight requirement per pair of cities

To meet the fundamental requirement:

• Morning and evening flights are required in both directions between each pair of cities.
• This means there are \( 4 \) flights connecting any pair of cities (2 each way, morning and evening).

Step 2: Number of unique city pairs

The number of ways to choose 2 cities from 10 cities is given by: \[ \binom{10}{2} = \frac{10 \times 9}{2} = 45 \]

Step 3: Total flights

Each of the 45 city pairs requires 4 flights: \[ \text{Total flights} = 45 \times 4 = 180 \]

Final Answer:

\[ \boxed{\text{Minimum number of flights} = 180} \]

Answer 2

Minimum Number of Flights: Hub–Non-Hub Model 

Step 1: City classification

We have 10 cities labeled A through J.

• Hubs: A, B, C (3 cities)
• Non-hubs: D, E, F, G, H, I, J (7 cities)

Rule: A direct flight must originate and/or terminate at a hub.

Step 2: Flights from a non-hub to hubs

Consider city D (non-hub). It must be connected to each of the 3 hubs (A, B, C). Given 4 flights between each pair of cities: \[ \text{Flights from D to hubs} = 4 \times 3 = 12 \]

Step 3: Apply to all non-hub cities

There are 7 non-hub cities. Each requires 12 flights to connect to all hubs: \[ \text{Flights (non-hub to hubs)} = 12 \times 7 = 84 \]

Step 4: Flights among hubs

The 3 hubs (A, B, C) must be interconnected. Number of unique hub–hub pairs: \[ \binom{3}{2} = 3 \] Each pair requires 4 flights: \[ \text{Flights (hub to hub)} = 3 \times 4 = 12 \]

Step 5: Total minimum flights

\[ \text{Total} = 84 \ (\text{non-hub to hub}) + 12 \ (\text{hub to hub}) \] \[ \boxed{\text{Minimum total flights} = 96} \]

Answer 3

Minimum Number of Direct Flights

Step 1: Group distribution

We have:

• G1: Cities A, B, C (3 cities)
• G2: 3 cities
• G3: 2 cities
• G4: 2 cities

Step 2: Flight restrictions

A city in G2 cannot have a direct flight to a city in G3 or G4. Thus, to travel from G2 to G3 or G4:

• All cities in G2 must connect to city A in G1.
• From A, passengers can travel to B or C, and then to cities in G3 or G4.

Step 3: Flights from G2 to A

Each city-to-city connection requires 4 flights (two in each direction, twice a day). Number of flights between the 3 cities in G2 and city A: \[ 3 \times 4 = 12 \ \text{flights} \]

Step 4: Flights from G3 to B

Each of the 2 cities in G3 connects to city B: \[ 2 \times 4 = 8 \ \text{flights} \]

Step 5: Flights from G4 to C

Each of the 2 cities in G4 connects to city C: \[ 2 \times 4 = 8 \ \text{flights} \]

Step 6: Interconnecting G1 cities

Cities A, B, and C must be fully connected: \[ \binom{3}{2} = 3 \ \text{pairs} \quad\Rightarrow\quad 3 \times 4 = 12 \ \text{flights} \]

Step 7: Total minimum flights

\[ \text{Total} = 12 \ (\text{G2–A}) + 8 \ (\text{G3–B}) + 8 \ (\text{G4–C}) + 12 \ (\text{within G1}) \] \[ \boxed{\text{Total flights} = 40} \]

Answer 4

Step 1: Initial information 

Cities in group G2 will be assigned to either G3 or G4. However, this change alone does not reduce the total number of flights, because cities in G2 must still remain connected to either city B or city C.

Step 2: Additional condition

An extra piece of information is given: There are now no flights between city A and city C.

Step 3: Impact of the change

In the previous setup, there were 4 scheduled flights between A and C. With the new restriction, these 4 flights will not be scheduled.

Step 4: Conclusion

Thus, the maximum reduction in the number of flights is: \[ \boxed{4} \]