A Senate committee has 5 Democrats and 5 Republicans. In how many distinguishable ways can they sit around a circular table if all the members of each party all sit next to each other?
As a block of 5 and 5 there are 10 ways to sit around the table 10.
Now , WITHIN EACH block there is 5! ways to sit
AND for EACH of the 5! ways to sit in one block there are 5! ways to sit in the OTHER block....
so 10 x 5! x 5! = 144,000 ways
(I am still learning this stuff like you.....as GA can attest! So this answer may be incorrect!) ~EP
Here is my solution, EP. Correct me if I'm wrong.....
Choose any 5 consecutive seats in which to place the Democrats -- it doesn't matter which 5 consecutive seats that we choose, since we can rotate the table. Then there are 5! ways to place the Democrats in their seats, and 5! ways to place the Republicans in their seats, for a total of 5!*5!=14,400 arrangements.
You may be correct.....I am assuming you can position the democrats and repub blocks 10 different ways araound the table....you are assuming there is only one way (it IS a round table after all).
That is why my answer is 10 times bigger than yours.....I hope someone who KNOWS the answer can chime in, but here is my reasoning (which may be incorrect for a ROUND table with no reference point.....but ok for a rectangular table....just dont know)
I suspect YOUR answer is correct as far as ORDER of the people go as you go around the table....but if the table is referenced as in my diagram to North...or the door of the room or the window in the room etc, then does the answer change?
What would happen if there were 11 chairs? (I think THAT would make my answer more correct)
Here's my take
Anchor the Republicans in any 5 seats.....note that it doesn't matter which 5 seats they occupy....if rotated, all of the arrangements will look the same......and they can be arranged in 5! = 120 ways
So.....the Democrats occupy the other 5 seats and they can be arranged in the same muber of ways = 5! = 120 ways
So...the total arrangemets are (Ways to arrange the Republicans) x ( Ways to arrange the Democrats) = 120 x 120 = 14,400
To see this..let's suppose that we have A,B,C,D and AB must sit as a block and CD must sit as a block
So we have 2! = 2 ways to arrange AB and 2! = 2 ways to arrange CD = 2! x 2! = 4 total arrangements
So.....the arrangements are
A A B B
C B D B C A D A
D C D C
Note that any rotations will still look the same with regards to order