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How many ways can three Biology majors, four Computer Science majors, four English majors, and two Physics majors sit at a round table, such that those in the same major sit together?

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  • $\begingroup$ Must all people of one major sit together? Or is just pairs of 2 okay? (For the even ones at least) $\endgroup$ – user12345 Apr 2 '17 at 5:12
  • $\begingroup$ @anonymaker000010001 For this question, all people in a certain major must sit with their major and no one else. $\endgroup$ – ProjectDefy Apr 4 '17 at 5:01
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I ended up finding the answer. I'll explain it as best as I possibly can on the subject I'm still having difficult times with.

So, this is a circular permutation type problem since it's a round table problem. So we have the equation so far:

$$ P_n=(n-1)! $$

Given our problem, we have $ P_n=(n-1)! = 3! $ number of ways of arranging the majors.

Now to account for the 3 Biology majors we have $3!$ ways to arrange them, 4 Comp. Sci Majors to be, $4!$ ways, 4 English Majors to be, $4!$ ways, and 2 Physics Majors to be, $2!$ ways. We have answer: $$ 3!*3!*4!*4!*2! = 41472 $$

Hence, to arrange three Biology majors, four Computer Science majors, four English majors, and two Physics majors sit at a round table, such that those in the same major sit together, would be $41472$ ways.

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    $\begingroup$ You should state that $P_n$ is the number of ways of arranging the majors around the table. $\endgroup$ – N. F. Taussig Apr 4 '17 at 8:09
  • $\begingroup$ @N.F.Taussig, I went ahead and added that where it seemed to fit! Thanks for the suggestion. $\endgroup$ – ProjectDefy Apr 5 '17 at 1:25
  • $\begingroup$ Your answer is correct. Another way to think about it is to seat a particular physics major. We use that person as our reference point. There are two ways to seat the other physics major (to the left or right of the first physics major to be seated). We can now seat the other three majors in $3!$ orders as we move clockwise around the circle. As you noted, there are $3!$ ways to arrange the biology majors, $4!$ ways to arrange the English majors, and $4!$ ways to arrange the Computer Science majors within their blocks, giving $2!3!3!4!4!$ possible seating arrangements at the round table. $\endgroup$ – N. F. Taussig Apr 5 '17 at 1:39
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Consider each group of people with the same major as 1 group. Then there are 4 groups and they can sit in 4! different ways, each group has people of the same major. Now, the group of biology major can sit in 3! different ways, the group of computer science major in 4! different way, etc. Thus the total come up to 4!3!4!4!2!

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  • $\begingroup$ I gave this a try and it still isn't correct (the answer for your equation was: 165888 ways). I did go ahead and try instead 4(3!4!4!2!) = 27648 ways, which was also incorrect. $\endgroup$ – ProjectDefy Apr 4 '17 at 5:00
  • $\begingroup$ I ended up finding the answer, which you can see posted if you were curious to find the answer. $\endgroup$ – ProjectDefy Apr 4 '17 at 5:19

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