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04 Jan 2010, 02:33
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E
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Difficulty:
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70% (00:37) correct
30%
(01:06)
wrong
based on 153
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Five noble knights are to be seated at a round table. How many different seating arrangements are possible, given that two seating arrangements are considered different only when the positions of the people are different relative to each other?
A. 120
B. 96
C. 60
D. 35
E. 24
M01-24
A. 120
B. 96
C. 60
D. 35
E. 24
M01-24
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08 Sep 2023, 08:46
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Five noble knights are to be seated at a round table. How many different seating arrangements are possible, given that two seating arrangements are considered different only when the positions of the people are different relative to each other. ?
A. 120
B. 96
C. 60
D. 35
E. 24
• The number of arrangements of \(n\) distinct objects in a row is given by \(n!\).
• The number of arrangements of \(n\) distinct objects in a circle is given by \((n-1)!\).
Why this difference? In a row, if you shift all objects by one position, you get a new arrangement. But in a circle, this shift results in the same relative arrangement. That's why the number of distinct arrangements for a circle is:
• \(\frac{n!}{n} = (n-1)!\).
For 5 knights, that is:
\((5-1)! = 4! = 24\).
Therefore, there are 24 different seating arrangements for the knights.
Answer: E
A. 120
B. 96
C. 60
D. 35
E. 24
• The number of arrangements of \(n\) distinct objects in a row is given by \(n!\).
• The number of arrangements of \(n\) distinct objects in a circle is given by \((n-1)!\).
Why this difference? In a row, if you shift all objects by one position, you get a new arrangement. But in a circle, this shift results in the same relative arrangement. That's why the number of distinct arrangements for a circle is:
• \(\frac{n!}{n} = (n-1)!\).
For 5 knights, that is:
\((5-1)! = 4! = 24\).
Therefore, there are 24 different seating arrangements for the knights.
Answer: E
General Discussion
21 Nov 2010, 21:06
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In case nothing is mentioned, in a circular arrangement, two seating arrangements are considered different only when the positions of the people are different relative to each other.
That is, in the question above, you will use the formula (n - 1)!
Here is the theory behind it:
Arranging 3 people (A, B, C) in a row: Following 6 ways
A B C, A C B, B A C, B C A, C A B, C B A
3! ways
Why is arranging 3 people in a circle different?
Look at one arrangement of 3 people A, B and C around a small round table O. Ignore the dots.
....A
....O
B......C
If I am B, A is to my left, C is to my right.
Look at this one now:
....C
....O
A......B
Here also, if I am B, A is to my left and C is to my right.
In a circle, these are considered a single arrangement because relative to each other, people are still sitting in the same position. This is the general rule in circular arrangement.
You use the formula n!/n = (n - 1)! because every n arrangements are considered a single arrangement. e.g. if n = 3, the given 3 arrangements are the same:
.....A ................ C ............... B
.....O ................ O .............. O
B........C ........ A ..... B ..... C........ A
In each of these, if I am B, I am sitting in the same position relative to others. A is to my left and C is to my right.
and these three are the same:
.....C ................ A ............... B
.....O ................ O .............. O
B........A ........ C ..... B ..... A........ C
Here, if I am B, C is to my left and A is to my right. Different from the first three.
Hence no. of arrangements = 3!/3 = 2 only
You might need to use n! in a circle if they mention that each seat in the circular arrangement is numbered and is hence different etc. Then there are just n distinct seats and n people. If nothing of the sorts is mentioned, you always use the (n - 1)! formula for circular arrangement.
That is, in the question above, you will use the formula (n - 1)!
Here is the theory behind it:
Arranging 3 people (A, B, C) in a row: Following 6 ways
A B C, A C B, B A C, B C A, C A B, C B A
3! ways
Why is arranging 3 people in a circle different?
Look at one arrangement of 3 people A, B and C around a small round table O. Ignore the dots.
....A
....O
B......C
If I am B, A is to my left, C is to my right.
Look at this one now:
....C
....O
A......B
Here also, if I am B, A is to my left and C is to my right.
In a circle, these are considered a single arrangement because relative to each other, people are still sitting in the same position. This is the general rule in circular arrangement.
You use the formula n!/n = (n - 1)! because every n arrangements are considered a single arrangement. e.g. if n = 3, the given 3 arrangements are the same:
.....A ................ C ............... B
.....O ................ O .............. O
B........C ........ A ..... B ..... C........ A
In each of these, if I am B, I am sitting in the same position relative to others. A is to my left and C is to my right.
and these three are the same:
.....C ................ A ............... B
.....O ................ O .............. O
B........A ........ C ..... B ..... A........ C
Here, if I am B, C is to my left and A is to my right. Different from the first three.
Hence no. of arrangements = 3!/3 = 2 only
You might need to use n! in a circle if they mention that each seat in the circular arrangement is numbered and is hence different etc. Then there are just n distinct seats and n people. If nothing of the sorts is mentioned, you always use the (n - 1)! formula for circular arrangement.
