# Thread: Permutations of a Mulit-Set

1. ## Permutations of a Mulit-Set

Determine the number of permutations of the multiset T = {3*a, 4*b, 2*c} where, for each type of letter, the letters of the same type do not all appear consecutively. (Thus aabbacbcb is allowed, but accbababb is not.)

I know the inclusion-exclusion principle is ideal for this situation, but I don't know how to do this..

Let $\mathcal{A}$ be the strings in which all a's are together.
That number is $\#\mathcal{A}$
Now count, $\#\mathcal{A}+\#\mathcal{B}+\#\mathcal{C}-\#\mathcal{A\&B}-\#\mathcal{A\&C}-\#\mathcal{B\& C}+\#\mathcal{A\& B\& C}$
I will get you started with $\#\mathcal{A}=\dfrac{7!}{4!\cdot 2!}$ and $\#\mathcal{A\& B}=\dfrac{4!}{ 2!}$