# Task 5. (12 points)

Asha is learning to write and can write three letters A, S, and Y. Her mother suggested she write seven letters in a row. In the resulting "word," three consecutive letters formed the name "ASY". How many such different seven-letter "words" exist?
# Solution.

We can consider the word ASYA as the fourth letter, then the seven-letter words will turn into five-letter words. The "letter" ASYA can stand in one of 5 positions, and in the other positions, any of the 3 letters can stand, so we get \(5 \cdot 3^{4} = 405\) words. Among them, there are words where the "letter" ASYA is written twice. These words are counted twice. Such words are \(3 \cdot 3 = 9\): ASYASYA, ASYAxASYA, ASYASYAx, where \(x\) is one of the letters A, S, or Y. Thus, the total number of different words containing the name ASYA is \(405 - 9 = 396\).

Answer: 396.

| Criterion Content | Evaluation | Points |
| :---: | :---: | :---: |
| Complete solution. | + | 12 |
| A minor computational error is made. | ![](https://cdn.mathpix.com/cropped/2024_05_06_8cbd0b9d96f75868c6e3g-5.jpg?height=65&width=192&top_left_y=1707&top_left_x=1577) | 10 |
| Calculations are correctly presented without accounting for the repetition of the name in the "word". The repetition of the name is considered, but errors are made. | $\pm$ | 9 |
| Calculations are correctly presented without accounting for the repetition of the name in the "word". The repetition of the name is not considered. | $+/ 2$ | 6 |
| Calculations are presented without accounting for the repetition of the name, some of which are correct. | 干 | 2 |
| The idea that the name can repeat is indicated. Calculations are incorrect. |  |  |
| The solution does not meet any of the criteria described above. | $-/ 0$ | 0 |
| Maximum score |  | 10 |