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Probability Level 4

In how many ways can a group of $8$ different guests (consisting of $4$ males and $4$ females) be seated at a round table with $8$ seats such that there are exactly $3$ males who are seated next to each other?

This is a part of the Set .

The answer is 1728.

2 solutions

Alan Yan
Aug 31, 2015

Thinking about how a table orientation would look like, it would look like three males together separated from another male by females.

Fix the lone male at the top of the table so we do not need to mind rotations.

There are $4$ ways to choose this male, and $3!$ ways to rearrange the three together males.

For the females, you can distribute them as one to the left and three to the right, two to the left, two to the right, three to the left and one to the right of the lone male, that is, $3$ possible orientations. There are $4!$ ways to rearrange the females.

Therefore, the total number of ways to sit is $4! \cdot 4! \cdot 3 = \boxed{1728}$

@Alan Yan ... But this a single case for the positions of the men and women . There can be other positions also.

Ayush Choubey - 5 years, 8 months ago

Clarify your argument. What do you not understand. I included all of the cases. List a case that I did not include.

Alan Yan - 5 years, 8 months ago

I have a confusion .. better solve it ... If I stand at a point nearby .. I could surely get 1728 arrangements . But if all the chairs are rotated ( clockwise or anticlockwise ) ; ( and all of them sitting on it ) ;; Won't it be a different arrangement . Mark my words .. I am saying I am on a fixed point.

Ayush Choubey - 5 years, 8 months ago

@Ayush Choubey – But that would still be the same arrangement -_- since you are only rotating the table. For the sake of an example let's say the table arrangement is BBBGGG where boys are indistinguishable and girls are indistinguishable. Then GBBBGG would be the same arrangement since it's a round table.

Alan Yan - 5 years, 8 months ago

Aaghaz Mahajan
Jun 8, 2018

Consider first the group of three males........they can be selected in four ways and can be permuted in 6 ways........hence, there are 24 ways for the three males to sit...
Next, place all the four women......this can be done in 24 ways..
Now, the remaining man will only have 3 slots remaining to sit...... ( those are between any two females ) and this can be done in 3 ways.........
So, the answer is 24 24 3 = 1728 ( Ramanujan's number - 1)....!!

Sit here

This is a part of the Set .

The answer is 1728.

2 solutions

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