Start off by splitting into two cases: the 7-rods connected at a vertex, and the 7-rods not connected.

With the 7-rods not connected:

There are only two possible tetrahedrons here- one with the 8-rods together, and one with the 8-rods across from each other. Any other combination is a rotation of one of these.

With the 7-rods connected:

In the face with the two 7-rods, the triangle could be finished with an 8-rod, or a 9-rod. With each of those possibilities, there will be different combinations for the other rods.

Hopefully this gets you started. I'll also tell you that your final answer should be less than 10.