Question: If a school's tuition plus room and board costs $50,000 per year, and they have 12 scholarships to give, does that mean they have $600,000 to split up for a team. Am I understanding the numbers correctly?           

Answer:  This is actually a fairly complex issue. While one answer might be yes, the Div. I school has $600,000 to divide among its 18-20 players, you must remember this is a total number, not an annual one. So even at a truly fully-funded college, the coach might only have $75,000 to spend in a given year.

Coaches can (and usually do) give partial amounts to different players, reserving the full ride---if they even give those---for major impact players such as pitchers. And if the coach graduates 4 athletes who are getting $30,000, $10000, $17000, and $18000 respectively, that coach might only be able to offer your player $15000 as a freshman because of what the coach is spending to replace other players in that group.

Also, this is assuming the school is fully funded with tuition and room and board.  Some schools have 12 scholarships total, but that might be 12 tuition scholarships only.  Their particular grants might not include room and board, leaving less money overall for recruiting. 

While a coach who has 12 scholarships might elect to give full rides to 10 players and divide the remaining 2 scholarships between 4 players, leaving the other kids on the team to fund their education in other ways, it's more likely he or she will chose to break up those scholarships.  If 18 players on the team are getting some amount of money, it's unlikely more than a couple of them are getting everything paid for.

Added note: It's also good to remember that many Div. I schools are not fully funded---i.e., they may have 4 or 7 or 9 scholarships to spend, and that Div. II schools are only allowed a total of just over 7 scholarships.