11.1. On the plate, there are various pancakes with three fillings: 2 with meat, 3 with cottage cheese, and 5 with strawberries. Svetlana ate them all sequentially, choosing each subsequent pancake at random. Find the probability that the first and last pancakes eaten were with the same filling.
Answer: $\frac{14}{45}$.

Solution: Two pancakes with the same filling can be either with meat, cottage cheese, or strawberries. Let's calculate the probabilities of each of these events and add them up. Arrange the pancakes in the order they are eaten. The probability that the first pancake is with meat is $\frac{2}{10}$. The probability that the last pancake is with meat is the same as the probability that the pancake with meat is in any other position. Therefore, this probability is $\frac{1}{9}$, since after choosing the first pancake, only 9 pancakes remain, one of which is with meat. Thus, the probability that the first and last pancakes are with meat is $\frac{2}{10} \cdot \frac{1}{9}=\frac{2}{90}$. Similarly, we find the probability that the first and last pancakes are with cottage cheese, which is $\frac{3}{10} \cdot \frac{2}{9}=\frac{6}{90}$. And the probability that the first and last pancakes are with strawberries is $\frac{5}{10} \cdot \frac{4}{9}=\frac{20}{90}$. Therefore, the answer to the problem is $\frac{2}{90}+\frac{6}{90}+\frac{20}{90}=\frac{28}{90}=\frac{14}{45}$.

Remark: The probability that the first and last pancakes are with meat can also be calculated as follows. There are $\frac{10!}{2!3!5!}$ ways to arrange our 10 pancakes in a row, and among them, there are $\frac{8}{3!5!}$ ways to arrange them in a row so that the pancakes with meat are at the beginning and the end. Then the probability that the first and last pancakes are with meat is

$$
\frac{\frac{8}{3!5!}}{\frac{10!}{2!3!5!}}=\frac{2}{90}=\frac{1}{45}
$$

Similarly, the probabilities for pancakes with other fillings can be calculated.

Comment: Any complete solution to the problem - 7 points. In a solution similar to the author's but containing arithmetic errors, the following criteria are summed up: the probability that the first and last pancakes are with meat is correctly found - 2 points; the probability that the first and last pancakes are with cottage cheese is correctly found - 2 points; the probability that the first and last pancakes are with strawberries is correctly found - 2 points; the correct answer is found - 1 point. Only the answer is provided - 0 points. The problem is not solved or solved incorrectly - 0 points.