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  • A council has 10 members, including the Chairman, Secretary, and Deputy Secretary. In how many ways can they be seated around a circular table if the Chairman and the Deputy Secretary must sit adja

A council has 10 members, including the Chairman, Secretary, and Deputy Secretary. In how many ways can they be seated around a circular table if the Chairman and the Deputy Secretary must sit adjacent to each other, but the Secretary cannot sit next to either of them?

 

Option: 1

45,360


Option: 2

18,780


Option: 3

36,258


Option: 4

60,124


Answers (1)

best_answer

To solve this problem, we can treat the Chairman and the Deputy Secretary as a single entity. This means that we have 9 entities (including the Secretary and the Chairman-Deputy Secretary combination) to arrange around a circular table.

The Secretary cannot sit next to either the Chairman or the Deputy Secretary. Let's consider the two cases separately:

Case 1: The Secretary sits between the Chairman and the Deputy Secretary

In this case, we can treat the Secretary, Chairman, and Deputy Secretary as a single entity. This means that we have 8 entities (including the Chairman-Deputy Secretary-Secretary combination) to arrange around a circular table.

The number of ways to arrange 8 entities in a circular arrangement is (8-1) !=7 !.
7 !=7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1=5,040

Case 2: The Secretary does not sit between the Chairman and the Deputy Secretary In this case, we have the Secretary, Chairman, and Deputy Secretary as separate entities. This means that we have 9 entities (including the Secretary, Chairman, and Deputy Secretary) to arrange around a circular table.
The number of ways to arrange 9 entities in a circular arrangement is (9-1) !=8 !.
8 !=8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1=40,320

Therefore, the total number of ways the 10 members can be seated around the circular table with the Chairman and the Deputy Secretary sitting adjacent to each other, but the Secretary not sitting next to either of them, is 7 !+8 !.

Note: The calculation assumes that the circular arrangement is considered distinct only when the relative positions of the members are different. If rotations are considered the same, we would need to divide the result by 10 (the number of members) to account for the different starting points.

So, the total number of ways the 10 members can be seated around the circular table with the Chairman and the Deputy Secretary sitting adjacent to each other, but the Secretary not sitting next to either of them, is 5,040+40,320=45,360.

Posted by

Ritika Harsh

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