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.
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
The Correct Option is C
Solution and Explanation
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:
\[ \boxed{\text{Minimum number of flights} = 180} \]
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
The Correct Option is C
Solution and Explanation
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
\[ \text{Total} = 84 \ (\text{non-hub to hub}) + 12 \ (\text{hub to hub}) \] \[ \boxed{\text{Minimum total flights} = 96} \]
Suppose the 10 cities are divided into 4 distinct groups 01,02,03,04 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
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
Correct Answer: 40
Solution and Explanation
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
\[ \text{Total} = 12 \ (\text{G2–A}) + 8 \ (\text{G3–B}) + 8 \ (\text{G4–C}) + 12 \ (\text{within G1}) \] \[ \boxed{\text{Total flights} = 40} \]
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?
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?
Correct Answer: 4
Solution and Explanation
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: \[ \boxed{4} \]
Top Questions on Conclusion
- A round table has seven chairs around it. The chairs are numbered 1 through 7 in a clockwise direction. Four friends, Aslam (A), Bashir (B), Chhavi (C), and Davies (D), sit on four of the chairs. In the starting position, Aslam and Chhavi are sitting next to each other, while for Bashir as well as Davies, there are empty chairs on either side of the chairs they are sitting on.
The friends take turns moving either clockwise or counterclockwise from their chair. The friend who has to move in a turn occupies the first empty chair in whichever direction they choose to move. Aslam moves first (Turn 1), followed by Bashir, Chhavi, and Davies (Turns 2, 3, and 4, respectively). Then Aslam moves again followed by Bashir, and Chhavi (Turns 5, 6, and 7, respectively).
The following information is known: 1. The four friends occupy adjacent chairs only at the end of Turn 2 and Turn 6. 2. Davies occupies Chair 2 after Turn 1 and Chair 4 after Turn 5, and Chhavi occupies Chair 7 after Turn 2. At InnovateX, six employees, Asha, Bunty, Chintu, Dolly, Eklavya, and Falguni, were split into two groups of three each: Elite led by Manager Kuku, and Novice led by Manager Lalu. At the end of each quarter, Kuku and Lalu handed out ratings to all members in their respective groups. In each group, each employee received a distinct integer rating from 1 to 3. & nbsp;
The score for an employee at the end of a quarter is defined as their cumulative rating from the beginning of the year. At the end of each quarter the employee in Novice with the highest score was promoted to Elite, and the employee in Elite with the minimum score was demoted to Novice. If there was a tie in scores, the employee with a higher rating in the latest quarter was ranked higher.
1. Asha, Bunty, and Chintu were in Elite at the beginning of Quarter 1. All of them were in Novice at the beginning of Quarter 4.
2. Dolly and Falguni were the only employees who got the same rating across all the quarters.
3. The following is known about ratings given by Lalu (Novice manager):
– Bunty received a rating of 1 in Quarter 2. & nbsp;
– Asha and Dolly received ratings of 1 and 2, respectively, in Quarter 3.- Faculty members in a management school can belong to one of four departments – Finance and Accounting (F&A), Marketing and Strategy (M&S), Operations and Quants (O&Q) and Behaviour and Human Resources (B&H). The numbers of faculty members in F&A, M&S, O&Q and B&H departments are 9, 7, 5 and 3 respectively. Prof. Pakrasi, Prof. Qureshi, Prof. Ramaswamy and Prof. Samuel are four members of the school's faculty who were candidates for the post of the Dean of the school. Only one of the candidates was from O&Q. Every faculty member, including the four candidates, voted for the post. In each department, all the faculty members who were not candidates voted for the same candidate. The rules for the election are listed below.
1. There cannot be more than two candidates from a single department.
2. A candidate cannot vote for himself/herself.
3. Faculty members cannot vote for a candidate from their own department.
After the election, it was observed that Prof. Pakrasi received 3 votes, Prof. Qureshi received 14 votes, Prof. Ramaswamy received 6 votes and Prof. Samuel received 1 vote. Prof. Pakrasi voted for Prof. Ramaswamy, Prof. Qureshi for Prof. Samuel, Prof. Ramaswamy for Prof. Qureshi and Prof. Samuel for Prof. Pakrasi - There are only four neighbourhoods in a city-Levmisto, Tyhrmisto,Pesmisto and Kitmisto. During the onset of a pandemic,the number of new cases of a disease in each of these neighbourhoods was recorded over a period of five days. On each day, the number of new cases recorded in any of the neighbourhoods was either 0,1,2 or 3.
The following facts are also known:
1. There was at least one new case in every neighbourhood on Day 1.
2. On each of the five days, there were more new cases in Kitmisto than in Pesmisto.
3. The number of new cases in the city in a day kept increasing during the five-day period. The number of new cases on Day 3 was exactly one more than that on Day 2.
4. The maximum number of new cases in a day in Pesmisto was 2 and this happenedonly once during the five-day period.
5. Kitmisto is the only place to have 3 new cases on Day 2.
6. The total numbers of new cases in Levmisto,Tyhrmisto,Pesmisto and Kitmisto over the five-day period were 12,12,5 and 14 respectively. - The Humanities department of a college is planning to organize eight seminars, one for each of the eight doctoral students - A, B, C, D, E, F, G and H. Four of them are from Economics, three from Sociology and one from Anthropology department. Each student is guided by one among P, Q, R, S and T. Two students are guided by each of P, R and T, while one student is guided by each of Q and S. Each student is guided by a guide belonging to their department.
Each seminar is to be scheduled in one of four consecutive 30-minute slots starting at 9:00 am, 9:30 am, 10:00 am and 10:30 am on the same day. More than one seminars can be scheduled in a slot, provided the guide is free. Only three rooms are available and hence at the most three seminars can be scheduled in a slot. Students who are guided by the same guide must be scheduled in consecutive slots.
The following additional facts are also known.
1. Seminars by students from Economics are scheduled in each of the four slots.
2. A’s is the only seminar that is scheduled at 10:00 am. A is guided by R.
3. F is an Anthropology student whose seminar is scheduled at 10:30 am.
4. The seminar of a Sociology student is scheduled at 9:00 am.
5. B and G are both Sociology students, whose seminars are scheduled in the same slot. The seminar of an Economics student, who is guided by T, is also scheduled in the same slot.
6. P, who is guiding both B and C, has students scheduled in the first two slots.
7. A and G are scheduled in two consecutive slots.
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