SOLUTION: how many was can 6 people be seated in a row if 3 of them insist on NOT sitting together?

Question 460272: how many was can 6 people be seated in a row if 3 of them insist on NOT sitting together?
Answer by Edwin McCravy(20055) (Show Source): You can put this solution on YOUR website!

I can't tell whether you mean that it is OK if two
of them sit together as long as the third one isn't
next to either of them, or whether you mean that no 
two of the three can sit together.

I will assume that it is OK if just two of them sit
together, but not all three.

There are 6! or 720 ways any of the 6 can sit anywhere.

Now we must subtract all the ways in which the three
sit together.  

There are 3! was they can sit in three adjacent seats.

When they sit together it is the same as seating 4 things
in order, because the three boys together is like a single
thing. So we must subtract 3!*4! from the 6!. So the answer is 

6! - 3!*4! = 720 = 6*24 = 720 - 144 = 576

It's a more complicated problem if no two of the three can
sit together.  The answer to that one is 144.  I won't show you
that one unless that was what you meant.

Edwin

RELATED QUESTIONS
How many ways can 6 people be seated in a row, if Ruth must be seated in the first chair? (answered by Edwin McCravy)

How many ways can 11 people be seated around a circular table if two of them insist to... (answered by ikleyn)

In how many ways can you arrange 4 boys and 4 girls in a row of 8 seats if one boy and a... (answered by josmiceli)

How many ways can four couples be seated in a row of eight seats? (a)if couples are... (answered by ewatrrr)

In how many ways may a group of 10 people be seated in a row with exactly 10 chairs if 2... (answered by sudhanshu_kmr)

In how many ways can 7 people be seated in a row of chairs if two of the people, David... (answered by robertb)

In how many ways can five students be seated in row of eight seat if a certain two... (answered by Edwin McCravy)

In how many ways can 7 people be seated in a row if two of them, Eddie and John, refuse... (answered by stanbon)

How many ways can seven people be seated in a row if two of them refuse to sit next to... (answered by stanbon)

if 5 people want to be seated, in how many ways can 4 of them be seated on a bench that... (answered by dfrazzetto)