Quoted from RonSS:So position 17 seems the culprit on the board. Replace the transistor (obviously test as well)?
Looks like it. With the machine off, you should never get 0 ohms continuity from any driver pin to ground. Since the solenoids are controlled by connecting them to ground, that means one would be on instantly when you turn the machine on.
pin 17 seems to be Q29 looking at the schematics online, which does correspond to the Plunger Kicker from the looks of it. Since that's the case, I'd replace the diode on that coil, and while it's off verify the resistance. Take the board out, test the transistor, if it doesn't read the same as its neighbors, replace it.
While you've got the board out, you should check if anything up the chain may be damaged. While Q29 is out, verify the two resistors that help drive it. They're part of RN5 and RN14, take a look on the schematics to find the right pin numbers, and make sure their resistance is right. Check that the associated capacitor (C63) isn't shorted. You can't test capacitors themselves, but you shouldn't get continuity across it. When in doubt, again compare the resistance of C63 with others.
You can also try to test U29 with the diode test of your DMM. There's a good guide on pinwiki but for some reason they've blocked google so I can't find it. Basically, put one lead on the ground pin (9) and one on each input and output (1 and 18 in this case), and compare with another set of pins (eg 2 and 17). Move the lead from ground to the high voltage pin (10) and try again. Reverse the leads and try again. As always, if something seems way off, you've probably got a bad part.
None of these tests are 100%, and they get less reliable the further in you go, but always worth a shot.