uh-oh, you used the word "copied" so I think I should use the word "licences"
As you are a school, you've got to keep yourself pretty clean and legal. So once you have chosen a game, you've got to obtain 20 legal copies of the game.
Then you've got to somehow remain legal whilst having them installed without a CD-Rom drive.
There are places you can get no-cd patches for them but I'm not sure how the licence issue is affected if you use them even when you have a legitimate copy/licence for each installation.
There are plenty of shareware games you can run that avoid most of this but you will most likely find that games for free or on the cheap aren't good enough to give the kids something to look forward to. Remember half of them will probably be more used to Halo etc. Somehow, a variation of solitaire doesn't quite compare.