Solving for an integer-valued triangle side lengths

Question

A triangle has sides of following dimensions $7cm$, $17cm$ and third side is an integer. Find the number of triangles possible.

What I did:- Maximum possible value of third side

As we know, third side will be surely less than the sum of the other two sides. $23<17+7$ Minimum possible value of third side Difference of any two sides is less than the third side $11>17-7=10$

Number of possible values between $11$ and $23$ are $13$. But the total number of triangles possible is $33$. How?

This is a gmat exam question.

Answer 1

It looks like you're right; there's probably an error with the answer.

Answer 2

Let $a$ be side-length of the third side.

Hence, $7+17>a$ and $7+a>17$, which gives $10<a<24$ or $11\leq a\leq 23$, which gives the answer: $13$.

I think your book is wrong.