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

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Solution

Number of direct flights within the group G1 (between A and B, B and C) = 12 - 4 = 8
Let’s consider all three flights in the group G2 are assigned to G3 hence increasing the number of flights in the group to 5 from 2. Note it doesn’t make any difference if you assign the flights to G3 or G4 as both groups have the same conditions.
Number of direct flights between B and any city in G3 $=^{5} C_{1} \times 4 = 20$
Number of direct flights between C and any city in G4 $=^{2} C_{1} \times 4 = 8$
Total number of flights after the operational difficulties arose = 8 + 20 + 8 = 36
Minimum number of direct flights to be scheduled before the operational difficulties arose = 12 + 12 + 8 + 8 = 40
Hence, Maximum reduction in the number of direct flights as compared to the situation before the operational difficulties arose = 40 - 36 = 4.

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