How many anagrams has the word KOMBINATOORIKA with the condition that the $3$ letters O can't be next to each other.
How to get to the answer?
My work. So the first thing I did, was to find how many anagrams the word "KOMBINATOORIKA" has in total, with no extra conditions I got $14! / (3! 2! 2! 2!)$ as an answer. Now I try to find how many different anagram contain combination "OOO". I could subtract answer from my first answer. That would give me end result. This is the idea.
On
Hint. Consider the total number of anagrams and subtract the number of anagrams where the three Os are adjacent.
P.S. According to your work the total number of anagrams is $$\frac{14!}{3!2!2!2!}$$ which is correct. Now we enumerate the anagrams where we consider the string OOO just as one letter. So the number of letters decreases from $14$ to $12$ and the number of such anagrams is $$\frac{12!}{2!2!2!}.$$ Finally we subtract this result from the first one: $$\frac{14!}{3!2!2!2!}-\frac{12!}{2!2!2!}.$$
Ok the solution is $\frac{14!}{2^3*6}-\frac{12!}{2^3}$.