How many ways can the letters a,a,a,a,b,b,b,b,c,c,c (4 a's, 4 b's, 3 c's) be arranged such that 4 consecutive letters are not the same?
There are a total of $\displaystyle \frac{11!}{(4!)^2(3!)}$ ways to arrange the string of letters
Now count number of ways that all a's or all b's are together.
Hint: $\displaystyle |A|+|B|-|A\cap B|$ then substract.