17. Given a wire of length $150 \mathrm{~cm}$, it is to be cut into $n(n>2)$ smaller segments, each of which has an integer length of no less than $1(\mathrm{~cm})$. If no three segments can form a triangle, find the maximum value of $n$, and how many ways there are to cut the wire into $n$ segments that satisfy the condition.
17. Since the sum of $n$ segments is a fixed value of $150(\mathrm{~cm})$, to make $n$ as large as possible, the length of each segment must be as small as possible. Given that the length of each segment is no less than $1(\mathrm{~cm})$, and no three segments can form a triangle, the lengths of these segments can only be $1,1,2,3,5,8$, $13,21,34,55,89, \cdots$.
But $1+1+2+\cdots+34+55=143<150$, 
hence the maximum value of $n$ is 10.
There are 7 ways to divide a wire of length $150(\mathrm{~cm})$ into 10 segments that meet the conditions:
$$
\begin{array}{l}
1,1,2,3,5,8,13,21,34,62 ; \\
1,1,2,3,5,8,13,21,35,61 ; \\
1,1,2,3,5,8,13,21,36,60 ; \\
1,1,2,3,5,8,13,21,37,59 ; \\
1,1,2,3,5,8,13,22,35,60 ; \\
1,1,2,3,5,8,13,22,36,59 ; \\
1,1,2,5,8,14,22,36,58
\end{array}
$$