2.1. [5-6.2 (20 points), 7-8.2 (15 points), 9.2 (15 points)] In how many ways can the word «РОТОР» be read from the letters in the diagram, if it is not allowed to return to the letters already passed, and readings that differ only in direction are considered the same?

| Р O Т O | дохо |
| :--- | :--- |
| O Т O P | О Х Д |
| Т O P | х О Д |
| O P | О Д |
| Р | Д |
Answer: 25.

Solution. The number of ways to read starting from the upper left letter "P" is $2^{4}=16$, as 4 steps are required to move through the letters, and each step is directed either to the right or down. If we strike out the considered letter "P", it remains to count the number of remaining ways in Fig. 1.

| O T O P | O T O | $\mathrm{T}$ | $\mathrm{T}$ |
| :---: | :---: | :---: | :---: |
| O T O P | O T O P | T O | $\mathrm{T}$ |
| T O P | T O P | T O P | T O |
| O P | O P | O P | O P |
| P Fig. 1 | $\mathrm{P}_{\text {Fig. } 2}$ | $\mathrm{P}_{\text {Fig. } 3}$ | $\mathrm{P}_{\text {Fig. }}$ |

Now, let's fix the reading direction: we will assume that the word is read such that the first letter "P" in it is not lower than the last. For the top right letter "P", we get 2 ways. Strike out this letter (Fig. 2) and consider the next letter "P". The number of ways to read starting from it is 3. Strike out this letter, as well as the letters "O" next to which no letter "P" remains, as there are no more ways to read containing these letters "O" (Fig. 3). The next letter "P" is in the same position as the previous one, so it also starts with 3 possible ways. Finally, strike it out and the "O" located above it (Fig. 4). Then only 1 way of reading remains.

In total, we get $16+2+3+3+1=25$ ways.