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?
[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
1.they sit anywhere they want
There are 8 positions around Mr. P.
Answer: 8! = 40320
2.Presidents sit together?
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
3.Secretaries sit together and treasurer's sit together?
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
