Question

there are 20 persons at a party. among the 20 people are 2 brothers.
1. the no of ways in which we can arrange them around a circle so that 2 brothers sit on either side of the host
2.the no of ways in which we can arrange them around a circle so that there is exactly 1 person between the 2 brothers

Answer 1

dear student

$$\begin{gathered} n\text{'}objectscanbearrangedaroundacirclein(n-1)! \\ Ifarrangingthese\text{'}n\text{'}objectsclockwiseorcounterclockwisemeansoneandthesame,thenthenumberarrangementswillbehalfthatnumber.i.e.,numberofarrangements=(n-1)!2 \\ Lettherebeexactlyonepersonbetweenthetwobrothersasstatedinthequestion. \\ Ifweconsiderthetwobrothersandthepersoninbetweenthebrothersasablock,thentherewill17othersandthisblockofthreepeopletobearrangedaroundacircle. \\ Thenumberofwaysofarranging18objectsaroundacircleisin17!ways. \\ Nowthebrotherscanbearrangedoneithersideofthepersonwhoisinbetweenthebrothersin2!ways. \\ Thepersonwhositsinbetweenthetwobrotherscouldbeanyofthe18inthegroupandcanbeselectedin18ways. \\ Therefore,thetotalnumberofways18\times 17!\times 2=2\times 18! \end{gathered}$$

regards