10. Fyodorovna's Tea Party. (7th grade. 3 points). Fyodorovna has 3 teacups. One day, she washed them and started a new life. Each time she wants to drink tea, Fyodorovna randomly picks one of the cups, and then puts it back with the others, but does not wash it again. Find the probability that during the fifth tea party since the start of her new life, Fyodorovna uses the last clean cup.
Solution. The specified event will occur only if the first four times Fedor used some two cups (with each being used at least once), and the fifth time - the last clean cup. For definiteness, let's number the cups. Then there are 16 ways to use cups 1 and 2 four times. Two of these ways are not suitable - these are the ways 1111 and 2222. Therefore, there are 14 ways to use two cups four times (each at least once). Similarly, there are 14 ways to use cups 1 and 3 four times, and another 14 ways to use cups 2 and 3 four times. Thus, there are 42 ways to use exactly two cups over four tea sessions and the third cup - on the fifth time. The total number of ways to use three cups five times is $3^{5}=243$. The desired probability is $\frac{42}{243}=\frac{14}{81}$.

Answer: $\frac{14}{81}$.