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A group of six boys and three girls went for a dinner after the show. They were given a round table for twelve people.

a) Find the number of possible arrangements if seats are not numbered and the group can be seated without restriction.
b) Find the number of possible arrangements if seats are numbered and the three girls have to sit together as a group, with empty seats to separate them from the boys.

jvdhooft
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Rachana Murali Narayanan
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1 Answers1

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  1. To seat the nine people at the table, choose nine seats and assign the people in a certain order. When the seats are unnumbered, one can simply divide the total number of valid arrangements by 12, so the number of arrangements is:

$${12 \choose 9} \cdot \frac{9!}{12}$$

  1. The three girls have to sit next to each other, so select a chair to seat one of the girls, and put two other girls next to her. If the boys are expected to sit together as a group, then either assign one empty chair to the left and two to the right, or two to the left and one to the right. Then, for the remaining chairs, arrange the six boys. In total, the number of arrangements is: $${12 \choose 1} \cdot 3! \cdot 2 \cdot 6!$$ If the boys do not have to sit together as a group, assign one empty chair to the left and one to the right of the girls, and arrange the six boys among the seven remaining chairs. In total, the number of arrangements is: $${12 \choose 1} \cdot 3! \cdot 7!$$
jvdhooft
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  • Thanks for the answer! – Rachana Murali Narayanan Jun 10 '17 at 08:30
  • @RachanaMuraliNarayanan You're welcome. If this or any other answer was of help to you, please consider upvoting or accepting. – jvdhooft Jun 10 '17 at 08:31
  • Your solutions make sense given your interpretation of the second question. However, the second question does not say that the boys have to sit together as a group, which raises the possibility that there is an empty seat among them. After placing an empty seat on each side of the girls, I would have selected one of the seven remaining seats as the remaining empty seat. – N. F. Taussig Jun 10 '17 at 23:22
  • @N.F.Taussig You are right, I updated my answer accordingly. – jvdhooft Jun 10 '17 at 23:44
  • Hi, I know this a bit late but may I know what the $${12 \choose 1} $$ refers to in the last $2$ answers? Are you fixing the girls' position? If so shouldn't it be $${10 \choose 1} $$ where we take $3$ seats as $1$ unit? – Leon Oct 30 '20 at 22:37
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    @Leon The seats are numbered, so we have to pick one chair to seat the leftmost girl (e.g., seat number 5). Then, the two other seats for the girls are determined (seats number 6 and 7, in this case). There are twelve possible chairs to start with, hence ${12 \choose 1}$. – jvdhooft Nov 02 '20 at 11:09