Which one of these guys are correct? If any. 2) In how many ways can a group of

Question

Which one of these guys are correct? If any. 2) In how many ways can a group of 8 children form a circle ifno pair of Mary, Betty and Alice can be next to each other? 2) In how many ways can a group of 8 children form a circle ifno pair of Mary, Betty and Alice can be next to each other? Answer: or Mary, Betty can sit together in 6! 2!ways Betty and Alice can sit together in 6!2!ways Mary, and Alice can sit together in 6!2!ways no pair of Mary, Betty and Alice can be next to eachother in 7! -3 (6!) (2!) ways=6! =120 ways Mary, Betty can sit together in 6! 2!ways Betty and Alice can sit together in 6!2!ways Mary, and Alice can sit together in 6!2!ways no pair of Mary, Betty and Alice can be next to eachother in 7! -3 (6!) (2!) ways=6! =120 ways

Explanation / Answer

2) In how manyways can a group of 8 children form a circle if no pair of Mary, Betty and Alice can be next to eachother? First consider seating the 5UNrestricted persons. There are (4!) ways to seat these 5 persons in a circle. Next, we add Mary to the circle, who we can place at any ofthe 5 "between person" positions.   There are now{(4!)*(5)} possible arrangements. We next add Betty to the circle, who can sit at any of the 4 {i.e., 6 - 2}"between person" places which are not next to Mary. We now have atotal of {(4!)*(5)*(4)} possible arrangements. Finally, we add Alice to the circle, who can sit at any of the 3 {i.e., 7- 4} "between person" places which are not next to eitherBetty or Alice. Hence, total number of possible arrangementsis {(4!)*(5)*(4)*(3)}: {Total # Arrangements}= (4!)*(5)*(4)*(3) =1440 .

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