Question

in a wedding party involving boys and girls how many ways can 12 boys and 8 girls be sitted at a round table if all the 8 girls do not sit together​

Answer 1

Answer:

Number of ways in which 4 married couples can be arranged around the circular table 

= (8-1)!      (∵ there are 8 persons)

= 7! = 5040   

(Note : If 8 persons are arranged in a linear way, there are 8! ways to do so. But when it come to a circular arrangement, the arrangements ABCDEFGH, BCDEFGHA, CDEFGHAB, DEFGHABC, EFGHABCD, FGHABCDE, GHABCDEF, HABCDEFG are same (due to rotation). i.e., for each of the 8! arrangements, there are 8 same arrangements. Hence we need to divide 8! by 8. i.e., total number of arrangements = 8!/8 = 7!. 

Another way to approach this : Arrange one person at first. Since it is a circular table, it does not make any difference where he sits. Other seats are then distinguishable relative to the occupied seat. Hence the remaining 7 can be arranged in 7! ways. Total number of ways = 7!

In general, n objects can be arranged in (n-1)! ways around a circular table.

Another aspect is when observations can be made from both sides like in the case of necklace. see the first question)