How many different arrangements are there of all the nine letters A, A, A, B, B, B, C, C, C in a row if no two of the same letters are adjacent?
First I tried to find how many ways to arrange so at least two similar letters are adjacent (the complementary) then subtract from total ways without restriction. I tried to do this via extended addition rule (i.e. with the three circle venn diagram) and now I'm confused how to calculate each case.
My attempt so far: let a be set of 'two A's adjacent to each other' let b be set of 'two B's adjacent to each other' let c be set of 'two C's adjacent to each other'
I need to find |complement of a U b U c| (Let Z be universal set, no restriction) = |Z| - |a| - |b| - |c| + |ab| + |bc| + |ca| - |abc|
I know |a| = |b| = |c| and |ab| = |bc| = |ca| therefore |complement of a U b U c| = |Z| - 3|a| + 3|ab| -|abc|.
|Z| = 9!/3!3!3!, but I'm not sure how to compute |a| or |ab| or |abc|