7.2. Daniil has 6 cards with letters, from which he managed to form the word WNMWNM shown in the picture. Note that this word has a remarkable property: if you rotate it 180 degrees, you get the same word. How many words with such a property can Daniil form using all 6 cards at once?

![](https://cdn.mathpix.com/cropped/2024_05_06_0f1f0fe01f8033fb92a1g-1.jpg?height=135&width=845&top_left_y=1326&top_left_x=637)
Answer: 12 words.

## Solution:

(1) According to the problem, Danil has 2 cards with the letter $\mathrm{N}$, which remains the same when flipped, and 4 cards with the letter M, which turns into the letter W when flipped. Clearly, to get a word with the desired properties, we need to arrange 2 letters M and one $\mathrm{N}$ in the first half, and the second half of the word will be uniquely determined.

(2) The letter $\mathrm{N}$ can be placed in three ways, after which the letter M can be placed on the first of the remaining positions in two ways (flip or not), and on the last position also in two ways. In total, there are $3 * 2 * 2=12$ options.

## Criteria:

Only the answer - 1 point.

Only the idea that the first three letters uniquely determine the word (part (1) of the solution) - 2 points (can be combined with the previous one).

After noting (1), listing all options with at least one missed case - no more than three points.

All 12 options are listed, but it is not explained why there are no other options - 1 point.