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Question
A round table conference is to be held among 20 delegates belonging from 20 different countries. The number of ways they can be seated
A round table conference is to be held among 20 delegates belonging from 20 different countries. The number of ways they can be seated
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Solution
The correct option is D when two particular delegates should always sit together is 2×18!
Number of ways to arrange n persons in circular table is (n−1)! ways
∴Total ways without any restriction is (20−1)!=19!
When two particular delegates should always sit together:
Let the two particular delegates who wish to sit together be treated as one unit. So we have 19 delegates who can be arranged on a round table in (19−1)!=18! ways.
After this, the two particular delegates can be permuted between themselves in 2!=2 ways.
Hence, number of required arrangements is =2×18!.
When two particular delegates should sit opposite to each other:
Let first delegate select the seat, it can be done in 1 way (because it is circular table)
Now second delegate can select the seat in 1 way opposite to first delegate.
Now the remaining persons can be arranged in 18! ways.
∴Total number of required ways when two particular delegates should never sit opposite to each other =19!−1×1×18!=18×18!
When there is exactly one person between two particular delegates:
Let the person be arranged in between two particular delegates in 18C1 ways.
The remaining arrangements can be done in (20−3+1−1)!=17! ways.
While two particular delegates can interchange among themselves.
∴Total ways
= 18C1×2!×17!=2×18!
∴ Total number of required ways when there is atleast two persons between two particular delegates is =19!−2×2×18!=15×18!
Number of ways to arrange n persons in circular table is (n−1)! ways
∴Total ways without any restriction is (20−1)!=19!
When two particular delegates should always sit together:
Let the two particular delegates who wish to sit together be treated as one unit. So we have 19 delegates who can be arranged on a round table in (19−1)!=18! ways.
After this, the two particular delegates can be permuted between themselves in 2!=2 ways.
Hence, number of required arrangements is =2×18!.
When two particular delegates should sit opposite to each other:
Let first delegate select the seat, it can be done in 1 way (because it is circular table)
Now second delegate can select the seat in 1 way opposite to first delegate.
Now the remaining persons can be arranged in 18! ways.
∴Total number of required ways when two particular delegates should never sit opposite to each other =19!−1×1×18!=18×18!
When there is exactly one person between two particular delegates:
Let the person be arranged in between two particular delegates in 18C1 ways.
The remaining arrangements can be done in (20−3+1−1)!=17! ways.
While two particular delegates can interchange among themselves.
∴Total ways
= 18C1×2!×17!=2×18!
∴ Total number of required ways when there is atleast two persons between two particular delegates is =19!−2×2×18!=15×18!
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