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there are 4 boys and 4 girls, how many ways they are arranged to sit in circular table if the 3 boys always together?

i found someone's video and the answer is just 5!3!, there are 4 boys why he didn't choose 3 boys from 4 boys? let say the 4 boys are A B C D and the girls E F G H why just count cyclic from ABC D E F G H. Can't it be A BCD E F G H ?

cahya python
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    Please edit for clarity. Are there $4$ boys or are there $3$? If there are $4$, what does it mean to refer to "the $3$ boys always together"? Do you mean that there is a specific group of $3$ boys that wants to sit together or do you mean that there must be a consecutive block of (any) three boys or more? Or something else? – lulu Feb 16 '24 at 17:03
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    Also, when dealing with circular arrangements, one sometimes says that rotations don't matter. Is that what you intended? – lulu Feb 16 '24 at 17:03
  • yes the rotations dont matter, and there are 4 boys. How to arrange them so that 3 out of 4 boys always together (sit next to each other) – cahya python Feb 16 '24 at 17:26
  • Again, is there a specific group of $3$ boys that must be sat together or are you just saying that we need some block of at least $3$ boys, no matter which? – lulu Feb 16 '24 at 17:48
  • specific group of 3 boys that must be sat together – cahya python Feb 16 '24 at 17:54

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The problem is very vaguely phrased, but reverse engineering from the solution:

Suppose there is a block of $3$ specific boys that must be sat together, call this block $X$ and say it consists of $B_1, B_2, B_3$. Then we must sit $G_1, G_2, G_3, G_4, B_4, X$, for which there are $6!$ possible arrangements. If we wish to disregard rotations, let's say that $G_1$ must always sit at the head of the table, so there are then $5!$ ways to arrange them. Now, $X$ can itself be sorted in $3!$ ways, so that makes the total $5!\times 3!$

To summarize: One interpretation of the question that matches the official result is "how many ways are there to place $4$ boys and $4$ girls at a round table given that there is a specified group of $3$ boys who must be seated in a block and given that arrangements that differ by a rotation are equivalent?"

Of course, different interpretations of the question will lead to different results.

lulu
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  • why the X just B1,B2, B3? cant it have other arrangement like B2,B3,B4 for X? i am confused because there are 4 choose 3 ways to determine the block called X – cahya python Feb 16 '24 at 18:53
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    Again: the question is vaguely phrased. As I have pointed out, it's not clear whether the questions wants a specific group of $3$ boys for $X$ or not. However, we have the official result, so we can reverse engineer it. If you read it as saying that the three boys in $X$ are fixed, then the answer matches, so I assume this is what was intended. If you want to work on the other problem, keep in mind that you must separately handle the case when all $4$ boys sit together. – lulu Feb 16 '24 at 18:55