Question 1088065
Three presidents,3 secretaries and 3 treasurer's are to 
occupy 9 seats for a round table discussion about a certain 
issue. In how many ways can they arranged themselves if
1.they sit anywhere they want
2.Presidents sit together?
3.Secretaries sit together and treasurer's sit together?
<pre>
[I hate round table problems (lol!) because they always make 
the very unrealistic assumption that the people, table and 
chairs were all sitting on a huge turntable, and that the 
huge turntable can be rotated so that regardless of how the
turntable is rotated, it's the same seating arrangement.]

The best way to handle these problems is to pick one of the 
n people to face north,  and then seat the other n-1 people 
around that person.  
Let's pick one of the presidents to face north, call him 
Mr. P.  Then we have 8 people composed of only 2 presidents, 
3 secretaries, and 3 treasurers to sit around Mr. P, like
this:

     P
  1     8
 2       7
  3     6
    4 5
</pre>
1.they sit anywhere they want
<pre>
There are 8 positions around Mr. P.

Answer: 8! = 40320
</pre>
2.Presidents sit together?
<pre>
We can place the 2 other presidents on Mr. P's right in 2! ways,
on his left in 2!, or one on each side of him in 2! ways.  That's 
3*2! ways to seat the other 2 presidents.  Then we can seat the 
other 6 people in 6! ways.

Answer 3*2!*6! = 4320 ways
</pre>
3.Secretaries sit together and treasurer's sit together?
<pre>
We can arrange the trio of secretaries in 3! ways.
We can arranges the trio of treasurers in 3! ways.  
Then we have 1 trio of secretaries, 1 trio of treasurers and 2
single people (presidents) to place around Mr. P.  That's 4 
"things" to place around Mr. P.  We can do that in 4! ways.

Answer: 3!3!4!  = 864 ways

Edwin</pre>