In the language of wolves has two letters $F$ and $P$, any finite sequence which forms a word.   А word $Y$ is called 'subpart' of word $X$ if Y is obtained from X by deleting some letters (for example,   the word  $FFPF$ has 8 'subpart's: F, P, FF, FP, PF, FFP, FPF, FFF).  Determine $n$ such that  the $n$ is the greatest number of 'subpart's can have n-letter word language of wolves.


F. Petrov,  V. Volkov
To determine the greatest number of 'subparts' that an \( n \)-letter word in the language of wolves can have, we need to understand the concept of subparts. A subpart of a word is any sequence that can be obtained by deleting some (or none) of the letters of the word without changing the order of the remaining letters.

1. **Understanding Subparts**:
   - For a word of length \( n \), each letter can either be included in a subpart or not.
   - Therefore, each letter has 2 choices: to be included or not included.

2. **Counting Subparts**:
   - For a word of length \( n \), the total number of possible subparts is given by the number of subsets of the set of \( n \) letters.
   - The number of subsets of a set with \( n \) elements is \( 2^n \).

3. **Excluding the Empty Subpart**:
   - The empty subpart (where no letters are included) is not considered a valid subpart.
   - Therefore, we need to subtract 1 from the total number of subsets to exclude the empty subpart.

4. **Calculation**:
   \[
   \text{Total number of subparts} = 2^n - 1
   \]

Thus, the greatest number of subparts that an \( n \)-letter word in the language of wolves can have is \( 2^n - 1 \).

The final answer is \( \boxed{ 2^n - 1 } \).