V10 has about 6 solenoids. One is attached to the starter as one unit the others are placed all over under the hood.
I experienced the same as you did. It was the solenoid mounted on the fender wall. :R
I had mobile repair out and he knew right away what the problem was. I turned the key as he checked out each solenoid until we found the one that wasn't working. Inexpensive part.
Just pray it is not the one attached to the starter because you have to buy the complete starter assembly for that one.
What happened to the good old days when vehicles had ONE starter solenoid. :B