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Eight guests have to be seated 4 on each side of a long rectangular table.2 particular guests desire to sit on one side of the table and 3 on the other side. The number of ways in which the sitting arrangements can be made is
- a)1732
- b)1728
- c)1730
- d)1278
Correct answer is option 'B'. Can you explain this answer?
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Most Upvoted Answer
Eight guests have to be seated 4 on each side of a long rectangular ta...
Solution:
Given: 8 guests, 4 on each side of a long rectangular table.
To be arranged: 2 particular guests on one side of the table and 3 on the other side.
Total ways to arrange 8 guests in 8 seats = 8! = 40320.
Since the table is rectangular, we have two sides and each side can accommodate 4 guests.
Number of ways to choose 2 guests from 8 guests = 8C2 = 28.
Number of ways to choose 3 guests from remaining 6 guests = 6C3 = 20.
Number of ways to arrange 4 guests on one side of the table = 4! = 24.
Number of ways to arrange 4 guests on the other side of the table = 4! = 24.
Therefore, the total number of ways to arrange the guests as per the given conditions = 28 * 20 * 24 * 24 = 172800.
However, we need to consider the cases where the 2 particular guests can switch sides.
Number of ways to arrange the guests with the 2 particular guests on one side = 172800/2 = 86400.
Hence, the required number of ways to arrange the guests as per the given conditions = 86400/2 = 17280.
Therefore, the correct option is B) 1728.
Given: 8 guests, 4 on each side of a long rectangular table.
To be arranged: 2 particular guests on one side of the table and 3 on the other side.
Total ways to arrange 8 guests in 8 seats = 8! = 40320.
Since the table is rectangular, we have two sides and each side can accommodate 4 guests.
Number of ways to choose 2 guests from 8 guests = 8C2 = 28.
Number of ways to choose 3 guests from remaining 6 guests = 6C3 = 20.
Number of ways to arrange 4 guests on one side of the table = 4! = 24.
Number of ways to arrange 4 guests on the other side of the table = 4! = 24.
Therefore, the total number of ways to arrange the guests as per the given conditions = 28 * 20 * 24 * 24 = 172800.
However, we need to consider the cases where the 2 particular guests can switch sides.
Number of ways to arrange the guests with the 2 particular guests on one side = 172800/2 = 86400.
Hence, the required number of ways to arrange the guests as per the given conditions = 86400/2 = 17280.
Therefore, the correct option is B) 1728.
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Community Answer
Eight guests have to be seated 4 on each side of a long rectangular ta...
Let the two particular guests sit on right side.
So the three particular guests will sit on left side.
So remaining will be 3 people which need to be selected.
From these 3 people 2 will sit on right side and the one will sit on left side.
Total ways of selecting the remaining will be = ³C₂ ⨯ ¹C₁ = 3
Total ways of arranging the people will be =
Selection of remaining × 4! ( for arranging people on left side) × 4! ( arranging people on right side)
= 3 × 24 × 24 = 3 × 576 = 1728
So in 1728 ways we can arrange them
So the three particular guests will sit on left side.
So remaining will be 3 people which need to be selected.
From these 3 people 2 will sit on right side and the one will sit on left side.
Total ways of selecting the remaining will be = ³C₂ ⨯ ¹C₁ = 3
Total ways of arranging the people will be =
Selection of remaining × 4! ( for arranging people on left side) × 4! ( arranging people on right side)
= 3 × 24 × 24 = 3 × 576 = 1728
So in 1728 ways we can arrange them
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Eight guests have to be seated 4 on each side of a long rectangular table.2 particular guests desire to sit on one side of the table and 3 on the other side. The number of ways in which the sitting arrangements can be made isa)1732b)1728c)1730d)1278Correct answer is option 'B'. Can you explain this answer?
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