We arrange 12-letter words having at our disposal five letters a, four letters b and three letters c. How many words are there without any block $5\times a$, $4\times b$ and $3\times c$. I need to use inclusion - exclusion principle. I counted all possible 12 letter words - $\frac{12!}{5!4!3!}$. Then words with block of letters:

only a blocks - $8\cdot \frac{7!}{4!3!}$

only b blocks - $9\cdot \frac{8!}{5!3!}$

only c blocks - $10\cdot \frac{9!}{5!4!}$.

Now I have to count all words, where block: a, b or a, c, or b, c appears together, but I don't now how.

only a blocks - $8\cdot \frac{7!}{4!3!}$

only b blocks - $9\cdot \frac{8!}{5!3!}$

only c blocks - $10\cdot \frac{9!}{5!4!}$.

Now I have to count all words, where block: a, b or a, c, or b, c appears together, but I don't now how.