SOLUTION: At a dinner party there are 10 people. They all sit at a round table. In how many ways can they sit if neither Amy nor Lucy want to sit next to Josh? I have tried this a few times

Question 1206743: At a dinner party there are 10 people. They all sit at a round table. In how many ways can they sit if neither Amy nor Lucy want to sit next to Josh? I have tried this a few times and get a different answer each time. Can you help please?
Answer by ikleyn(52778) (Show Source): You can put this solution on YOUR website!
.
At a dinner party there are 10 people. They all sit at a round table.
In how many ways can they sit if neither Amy nor Lucy want to sit next to Josh?
I have tried this a few times and get a different answer each time. Can you help please?
~~~~~~~~~~~~~~~~~~~~~~~~

In such problems, the indistinguishable arrangements are those that are obtained 
one from the other by a circular rotation. 


For 10 people around a round table, there are 9! distinguishable arrangements 
(= a universal set U of arrangements).

Of them, there are 8! arrangements, where A sits next to Josh on the left of him
and 8! arrangements, where A sits next to Josh on the right of him.


So, in the universal set U, there are 2*8! arrangements {A}, where A sits next to Josh.


Similarly, in the universal set U, there are 2*8! arrangements {L}, where L sits next to Josh.


These arrangements, {A} and {L}, have non-empty intersection.


This intersection has 7! arrangements {AJL} and 7! other arrangements {LJA}.


From it, we conclude that the number of favorable arrangements is 

    9! - 2*8! - 2*8! + 2*7!  = 1*2*3*4*5*6*7*8*9 - 4*(1*2*3*4*5*6*7*8) + 2*(1*2*3*4*5*6*7) = 211680.    ANSWER

Solved.

RELATED QUESTIONS
There are 2 boys and 5 girls at a party. In how many ways can they sit at a round table,... (answered by Edwin Parker)

in how many ways can 7 people be seated at a round table if they can sit anywhere? (answered by Jc0110)

9 people are sitting a round a table. How many ways can they... (answered by stanbon)

In how many ways can 7 people be seated at a round table if they can sit anywhere and 2... (answered by Jc0110)

how many ways may 6 people be sealed at a round table such that they can still sit... (answered by richwmiller)

A group of 3 English majors, 2 anthropology majors, and 5 history majors are going out to (answered by Edwin McCravy)

There are 6 places to sit at a table. How many ways can 6 people be seated at the table? (answered by stanbon)

5 people are to be ined up for a dinner. How many ways can they stand in a line? How... (answered by stanbon)

there are 3 boys and three girls at a party. in how many ways can they be seated in a row (answered by josmiceli)