The only way to get $x^{17}$ is two $x^5$ and one $x^7$. Thus,

• We first select one out of $20$ terms to contribute the $x^{7}$ term. There are $20$ choices.
• Next, we choose two out of $19$ terms to contribute the $x^5$ term. There are ${19\choose 2}=171$ choices

Finally, by the principle of multiplication, we have, the answer is $$20 \times 171=\boxed{3420}$$