3. (3 points) From cards with letters, you can form the word KARAKATITSA. How many words (not necessarily meaningful) can be formed from these cards, in which the letters R and T are adjacent?
Answer: $\frac{9!}{4!}=9 \cdot 8 \cdot 7 \cdot 6 \cdot 5=15120$

## Solution:

Let's replace the adjacent letters R and T with one letter, for example, the letter Sh, since they stand together. Then we are left with 9 letters, and among them, 4 letters A and 2 letters K.

Thus, the number of ways to rearrange the letters in this word is $\frac{9!}{4! \cdot 2!}$. Additionally, the letter Sh can be decoded in two ways: as RT and as TR, which doubles the number of ways. Therefore, we get $\frac{9!}{4!}$.