You may or may not like the following solution, which uses exponential generating functions.
Let's say, more generally, we want to find the number of ways to select r balls from the set of cricket, tennis, and squash balls. Call this number .
Let f be the exponential generating function of , i.e.
by elementary properties of exponential generating functions. (It's easy to see once you know how.) The answer to your problem is , which is the coefficient of when f is expanded.
The bad news is that finding the coefficient by pencil and paper essentially boils down to listing the cases, which you said you want to avoid. But the good news is that there are many computer algebra systems (Wolfram Alpha, for one) which will expand f for you. When this is done we find
as you predicted.