9.5. On a shelf, there are 12 books. In how many ways can 5 of them be chosen if books standing next to each other cannot be chosen? Justify your answer.
Solution: Let's place one more, the thirteenth book, on the right side of the shelf. We will mentally glue each selected book to the book to its right. This is possible since we do not select the 13th book and do not select adjacent books. This will result in 5 glued two-volume sets and 3 separate books. Therefore, the number of ways is the same as the number of ways to arrange 5 two-volume sets (considering them identical) and 3 single-volume books (also identical) in a row. This number is equal to the number of combinations of 8 taken 3 at a time, which is $\frac{8!}{3!(8-3)!}=56$.

Note: Other methods of solving are possible (induction by the number of books, direct enumeration, etc.).

Answer: 56 ways.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| A correct approach but an incorrect answer due to arithmetic errors | 6 points |
| The correct idea of "gluing" a book to the one on its right did not lead to the correct answer because the rightmost book was not correctly accounted for | 4 points |
| In solving by complete enumeration, not all cases were considered | 2 points |
| Correct answer without justification | 1 point |