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16 Jul 2023, 12:15
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
51% (01:12) correct
49%
(01:17)
wrong
based on 470
sessions
History
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Five women are to be seated around a table, and then each to five men will be assigned a position behind a women. In how many ways can couples thus be arranged, if only the relative order around the table is importance?
A. 2*5!
B. 5!*5!
C. 4!*5!
D. 4!*4!
E. 10!
A. 2*5!
B. 5!*5!
C. 4!*5!
D. 4!*4!
E. 10!
04 Aug 2023, 19:41
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stepanyan
I like to translate round table problems into linear problems first.
So, let's forget that there was round table and find how many linear arrangements are possible.
Suppose there is a circular arrangement ABCDE in clockwise order.
Now take these 5 people ABCDE and arrange them in a row.
1) A....B....C....D....E
Now notice the following arrangements:
2) E...A....B....C....D
3) D...E...A....B....C
4) C...D...E...A....B
5) B...C...D...E...A
In all these arrangements, I have just pushed every term to the left and transferred the last term of every arrangement to the first term of the next arrangement.
Now try to arrange each of the above 5 arrangements around a round table. You will notice that you will get an identical circular arrangement every time.
Therefore, for every circular arrangement of items ABCDE, there exists 5 linear arrangements that are identical when translated to circular.
Basically it means, you can ignore the circular nature of a problem and just assume the problem to be linear. Then you can translate it back to a circular problem by dividing the result of the linear problem by the number of items being arranged.
Mathematically represented as:
Number of linear arrangements of n different things = n X (number of circular arrangements of n different things)
Now let's solve this problem:
Suppose there are 5 boxes arranged in a row.
B1...B2....B3....B4....B5
You have to distribute 5 men and 5 women in those boxes. Each box will have 1 man and 1 woman.
No. of ways to arrange 5 men in 5 boxes = 5!
No. of ways to arrange 5 women in 5 boxes = 5!
So, the number of unique ways to fill the boxes = 5!*5!
Now let's arrange the boxes in a circle and eliminate the duplicate cases.
Using our formula:
Number of linear arrangements of n different things = n X (number of circular arrangements of n different things)
Here,
No. of linear arrangements of boxes = 5!*5!
n=5
Therefore, no. of circular arrangements of the boxes = (5!*5!)/5 = 4!*5!
Option C is correct.
General Discussion
05 Aug 2023, 01:24
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stepanyan
Assume the women are W1, W2, W3, W4 and W5 and the seats around the table are S1, S2, S3, S4 and S5. Since only relative order is important, the following two orders would be considered the same.
S1: W1
S2: W2
S3: W3
S4: W4
S5: W5
S1: W5
S2: W1
S3: W2
S4: W3
S5: W4
In both the cases, W5 is to the left of W1 and W2 is to the right of W1 and so on. So lets fix the position of one of the women. Now the other 4 women can be arranged in the other 4 seats in any order. Hence, the women can be arranged in 4! ways.
Once the women are arranged, the men can be arranged in 5! ways behind them since any man can occupy any of the five spaces available.
Hence the answer is 4!*5!, option C
