Puzzle #105: Play Ball!

In one inning, the home team had all nine players at bat. The team got two singles, two doubles, one triple, one home run, and three outs at bat.

Assume that for each hit, all players already on base advance the same number of bases as the hit was for (until they reach home plate). What is the maximum number of runs that the team could have scored? Why? Give an arrangement of the batting above that shows the maximum.

Now, do the same assuming that for each hit, all players already on base advance the minimum amount they can (while still advancing).

Is there a difference in the maximums? Why or why not?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, Jun 10, 2015 at noon Pacific Time. I will post the answer shortly after.