Question

The panel for a political forum consists of four Democrats, 5 republicans, 3 independents in how many orders can they be seated if all that matters is party not individual

Answer 1

Your answer is 8 so good luck

Answer 2

I'm assuming we can say each Democrat, Republican, and Independent are the same element.

Since there are 12 seats, we can seat them in 12! different ways, assuming straight line.
$\text{Total(no restrictions): } 12!$

However, the four democrats are the same element, since we only care about the different parties. Using this same logic, the 5 Republicans can swap without us knowing and the 3 Independents, as well.

Thus, we've overcounted by a factor of 4! 5! 3!, because each element are identical.

To counter this, we divide by the number of elements repeated:
$\text{Total ways: } \frac{12!}{4!5!3!} = 27,720$