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Could someone please help me with this, I am getting an answer 240. But I'm not sure if I'm correct.

Find the number of ways in which four men, two women and a child can sit at a table if the child is seated between two women.

Regards, Subhash

Note: 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 \((n-1)!\).

We have M, M, M, M, W, W, C --> glue two women and the child so that they become one unit and the child is between women: {WCW}. Now, these 5 units: {M}, {M}, {M}, {M}, {WCW} can be arranged around the table in (5-1)!=4! ways and the women within their unit can be arranged in 2 ways {W1, C, W2} or {W2, C, W1} so total # of arrangement is 4!*2=48.

Answer: 48.

240 would be the answer if the arrangement were in a row: {M}, {M}, {M}, {M}, {WCW} can be arranged in a row in 5! ways and the women within their unit can be arranged in 2 ways {W1, C, W2} or {W2, C, W1} so total # of arrangement is 5!*2=240.

Thanks for your reply. And I did not have the answer, so could not post it, apologies.

One question, if the places in table are numbered, wouldn't it become a case like arranging the members in the stated manner in a line, in which case the answer 240 is valid ?

Regards, Subhash
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Thanks for your reply. And I did not have the answer, so could not post it, apologies.

One question, if the places in table are numbered, wouldn't it become a case like arranging the members in the stated manner in a line, in which case the answer 240 is valid ?

Regards, Subhash

If the chairs are numbered and one specific arrangement and the same arrangement but shifted by one position are considered different then the answer will simply be 48*7.

That's because the difference between placement in a row and that in a circle is following: if we shift all object by one position, we will get different arrangement in a row but the same relative arrangement in a circle.
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240 would be the answer if the arrangement were in a row: {M}, {M}, {M}, {M}, {WCW} can be arranged in a row in 5! ways and the women within their unit can be arranged in 2 ways {W1, C, W2} or {W2, C, W1} so total # of arrangement is 5!*2=240.

For this szenario I received 744 different cases, including the cases when the child and a man or men are between the 2 women.

{M}, {M}, {M},{WCMW} 4*3*2*2*2=96 {M}, {M},{WCMMW} 3*2*3*2=72 {M},{WCMMMW}2*4*3*2*2=96 {WCMMMMW}5*4*3*2*2=240 These possibilities together with Bunuels possibilities when only the child is btw the women {WCW} gives 504+240=744

240 would be the answer if the arrangement were in a row: {M}, {M}, {M}, {M}, {WCW} can be arranged in a row in 5! ways and the women within their unit can be arranged in 2 ways {W1, C, W2} or {W2, C, W1} so total # of arrangement is 5!*2=240.

For this szenario I received 744 different cases, including the cases when the child and a man or men are between the 2 women.

{M}, {M}, {M},{WCMW} 4*3*2*2*2=96 {M}, {M},{WCMMW} 3*2*3*2=72 {M},{WCMMMW}2*4*3*2*2=96 {WCMMMMW}5*4*3*2*2=240 These possibilities together with Bunuels possibilities when only the child is btw the women {WCW} gives 504+240=744

Please correct me if I am wrong!

I think the question means that ONLY the child must be between two women.
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Could someone please help me with this, I am getting an answer 240. But I'm not sure if I'm correct.

Find the number of ways in which four men, two women and a child can sit at a table if the child is seated between two women.

Regards, Subhash

Note: 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 \((n-1)!\).

We have M, M, M, M, W, W, C --> glue two women and the child so that they become one unit and the child is between women: {WCW}. Now, these 5 units: {M}, {M}, {M}, {M}, {WCW} can be arranged around the table in (5-1)!=4! ways and the women within their unit can be arranged in 2 ways {W1, C, W2} or {W2, C, W1} so total # of arrangement is 4!*2=48.

Answer: 48.

240 would be the answer if the arrangement were in a row: {M}, {M}, {M}, {M}, {WCW} can be arranged in a row in 5! ways and the women within their unit can be arranged in 2 ways {W1, C, W2} or {W2, C, W1} so total # of arrangement is 5!*2=240.

Another way to look at the question. (though let me point out first that the question doesn't specifically say 'circular table'. It just says 'sit at a table'. If it is a rectangular table, perhaps there are 4 chairs on one side, 3 on the other etc. Since you mentioned "Circular Permutation Problem" in the subject line, I am assuming it is meant to be a circular table.)

7 seats around a circular table, 7 people. First I make the child sit anywhere in 1 way since all seats are the same. The two women can sit around him in 2! ways. Now 4 seats are left for 4 men and they can occupy them in 4! ways. Total number of ways = 4!*2! = 48

Another thing, if the places are numbered, say 1, 2, 3 etc for the 7 seats, the number of arrangements will be 7*2!*4! = 336. Make the child sit on any one of the 7 seats since all are unique now. The women sit around the child in 2! ways and the men sit on the rest of the 4 seats in 4! ways. The reason why this number is greater than the number of arrangements in case of a row (240 ways) is because in a row, child cannot be in 1st or 7th position while in a circle, the child can sit on seat no 1 or seat no 7. So we have 2*2!*4! = 96 extra cases in case of numbered seats around a circular table. Note: 240 + 96 = 336
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