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:

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120

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Solution

The correct option is C
96

Number of hub cities = 3
Number of direct flights among the hub cities $=^{3} C_{2} \times 4 = 12$ Number of direct flights from 3 hub cities to 7 non-hub cities $=^{3} C_{1} \times^{7} C_{1} \times 4 = 84$
Total number of direct flights = 84 + 12 = 96

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Similar questions

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 (not-stop) flights, then the minimum number of direct flights to be scheduled 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
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Q. 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?

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The correct option is C 96 Number of hub cities = 3 Number of direct flights among the hub cities =3C2×4=12 Number of direct flights from 3 hub cities to 7 non-hub cities =3C1×7C1×4=84 Total number of direct flights = 84 + 12 = 96

The correct option is C
96

Number of hub cities = 3
Number of direct flights among the hub cities $=^{3} C_{2} \times 4 = 12$ Number of direct flights from 3 hub cities to 7 non-hub cities $=^{3} C_{1} \times^{7} C_{1} \times 4 = 84$
Total number of direct flights = 84 + 12 = 96