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In how many ways can 4 arts students and 4 science students

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In how many ways can 4 arts students and 4 science students [#permalink]  22 Feb 2004, 09:30
In how many ways can 4 arts students and 4 science students be arranged alternately round the table.
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Paul

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Assuming that the students are distingushable,
then it is 7!.
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XAXAXAXAX- A can sit in 3! and S can occupy 5 places or 4! ways for S students seating, so i would go with 3!x4!=144.
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Assuming that the students are distingushable,
then it is 7!.

It is either 4!*3! or 7*4!*3! . But I feel it is 7*4!*3!
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Explanation:
Assuming A_B_C_D_ for art students positions and _E_F_G_H for science students. There are (4-1)! ways of sitting art students around the table and (4-1)! ways of sitting science students around the table. Also, there are 4C1 way for any given group's student to sit between any member of the other group. For instance, E could be between A/B or between B/C or between C/D or right after D. Therefore, answer is 3! * 3! * 4C1 = 144. Good job guys!
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Paul wrote:
Explanation:
Assuming A_B_C_D_ for art students positions and _E_F_G_H for science students. There are (4-1)! ways of sitting art students around the table and (4-1)! ways of sitting science students around the table. Also, there are 4C1 way for any given group's student to sit between any member of the other group. For instance, E could be between A/B or between B/C or between C/D or right after D. Therefore, answer is 3! * 3! * 4C1 = 144. Good job guys!

I agree with your answer but the explanation does not make sense.

Let's put student A as our point of reference and anchor her to the 1st position on the table. There are now 3 available positions at the table specifically for art sutdetns 4 avaibable position for the science student. Hence, there are 3! x 4! difference distinct arrangements of the "kichen sind athors)
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AkamaiBrah
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MBA, Anderson School of Management, UCLA, Class of 1993

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AkamaiBrah wrote:
I agree with your answer but the explanation does not make sense.

Let's put student A as our point of reference and anchor her to the 1st position on the table. There are now 3 available positions at the table specifically for art sutdetns 4 avaibable position for the science student. Hence, there are 3! x 4! difference distinct arrangements of the "kichen sind athors)

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4 Arts and 4 Science
3!*2! * 3!*2!

Ways to sit 4 Arts students is (4-1) = 3!. Now you can have this in 2 ways
so total is 3!2! = 12
same for science = 12

total = 12*12=144

I have multiplied 2! in both cases because both set of students have two seperate arrangements.

One which starts with arts student: A S A S A S A S
and One which starts with science student: S A S A S A S A

so both groups of arts and science students can be seated in 2! ways respectively
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pakoo wrote:
4 Arts and 4 Science
3!*2! * 3!*2!

Ways to sit 4 Arts students is (4-1) = 3!. Now you can have this in 2 ways
so total is 3!2! = 12
same for science = 12

total = 12*12=144

I have multiplied 2! in both cases because both set of students have two seperate arrangements.

One which starts with arts student: A S A S A S A S
and One which starts with science student: S A S A S A S A

so both groups of arts and science students can be seated in 2! ways respectively

I'm sorry, but your explanation does not make sense to me.
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AkamaiBrah
Former Senior Instructor, Manhattan GMAT and VeritasPrep
Vice President, Midtown NYC Investment Bank, Structured Finance IT
MFE, Haas School of Business, UC Berkeley, Class of 2005
MBA, Anderson School of Management, UCLA, Class of 1993

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Thanks for the solution AkamaiBrah. I understand what you mean.

Consider just the arts students first. They can be seated in a circle in (4-1)! = 3!

Now there are 4 empty places in between each Arts students and each of these empty places needs to be filled with a science student. 4 places and 4 science students. Thus, 4! ways

Total no. of ways = 3! x 4! = 144 (Answer)
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