A panel for a political forum is made up of 12 people from three parties, all seated in a row. The panel consists of 5 Democrats, 1 Green Party member, and 6 Libertarians. In how many distinct orders can they be seated if two people of the same party are considered identical (not distinct)?

Step-by-step explanation:

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.

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:

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.