Question

In how many ways can 12
people be seated around a rectangular
table with 4 seats each on two opposite
sides and 3 seats each on the other
two sides?
​

Answer 1

Answer:

7*13!

Step-by-step explanation:

please update the question

there should be 14 total people. 4 on two opposite side makes it 8 people and 3 on each side makes it 6 people.

total = 8+6 = 14 people. i m providing answer considering 14 people

First try to understand why is the number of arrangements of 14 people in a circular table is 13! and not 14!

Before anyone sits, each seat in the table is symmetrical to each other.

So, the first person, wherever he/she sits, it's the same thing. Thus, the first person can be seated in only 1 way. Now, once this person is seated, all the seats become unsymmetrical w.r.t. this person.

Hence, the remaining 13 can be seated in 13! ways.

In this problem, there is a rectangular table having 4 seats one the longer sides and 3 on the shorter sides.

First draw the rectangle and mark seat numbers from 1 to 14, clockwise or counter-clockwise. Once you're done, you'll find that seat number 1 and seat number 8 are symmetrical to each other. Seat number 2 and seat number 9 are symmetrical to each other, and so on till the seats pair 7,14. Just rotate the rectangle by 180 degrees and you'll find the symmetrical pair of seats exchanging places.

Thus, there are 7 pairs of symmetrical seats. But, a seat of any given pair is unsymmetrical to a seat of any other pair. For example, 1&2 are unsymmetrical, 1&3 are, etc..

Therefore, here the first person can be seated in 7 different ways (i.e. one of the 7 unsymmetrical seats). Once the first person is seated, now again all seats become unsymmetrical. So, the remaining 13people can be now seated in 13! ways.

So, total ways =7*13!