Hi. I'm struggling with this question from the section on the inclusion/exclusion principle:

"How many arrangements of the numbers 1, 1, 2, 2, 3, 3, 4, 4 are there in which no adjacent numbers are equal?"

My thought was to try and find the total arrangements possible, then subtract off the number of arrangements with at least 2 numbers equal, the latter being calculated with the help of inclusion/exclusion.

Calculating the total arrangements seems easy: C(8,2)*C(6,2)*C(4,2)*C(2,2)

But the other part seems complicated. My first attempt was to divide the eight slots into 7 pairs of adjacent slots. Then A_x is the number of arrangments that make pair x equal. But then, I have to go seven levels deep in applying inclusion/exclusion, and I'm doing about half a page of careful case calculation before even finishing n_2. When a counting problem gets highly complicated and calculation intensive, that usually means I'm taking the wrong approach.