What is the number of anagrams of the word AAABBBCDEF such that no three consecutive letters are identical
Answer: 88080. First, let's find the total number of anagrams. There are $10 \times 9 \times 8 / 6$ ways to place the letters $A$, then $7 \times 6 \times 5 / 6$ ways to place the letters $B$, and finally $4 \times 3 \times 2$ ways to place the other letters, making a total of 100800.

Similarly, we count that there are 6720 anagrams such that the $A$s are consecutive, and the same for $B$s. Therefore, the number we are looking for is $100800-6720 \times 2+x$ where $x$ is the number of anagrams such that both the $A$s and the $B$s are consecutive. Now, $x=2 y$ where $y$ is the number of anagrams such that the $A$s are consecutive, the $B$s are consecutive, and the $B$s are after the $A$s. We count that $y=(5+4+3+2+1) \times(4 \times 3 \times 2 \times 1)=360$, so the final answer is 88080.