Yep. You're right. Only 5 computers can be authorised (at the same time) to one iTunes account.
Some people read this as one iTunes account to five devices (iPads, etc...). The actual Terms and Conditions refer to "five iTunes authorised devices". You do not need to authorise an iOS device.
This is a complicated area, but it is pretty clear that you cannot buy an app and then sync it to a class set of devices that you use at your school. I have heard that in Australia, changes in Terms and Conditions are not applied in retrospect (so if you bought an app when the Terms and Conditions stated one thing, then those T&C apply to that product until you update it, and purchase it under newer T&C). I'm not sure if this is true, though.