Let's say you have a set of order statistics $ X_{(1)}, \dots, X_{(N)} $ drawn from a discrete uniform distribution $ \text{unif}(1,S) $. If you choose $ X_{(n_1)}, \dots, X_{(n_k)} $ from this set, how would you find the distribution $ Y = \sum_{j=1}^k X_{(n_k)} $? As an example to make it clearer, I've been trying to develop a tabletop RPG system and I'm thinking that the player could level up their stats in a few different ways. One way I'm imagining is that players can roll--for example--4 d6's and then must start off with $ X_{(1)} + X_{(2)} $, then being able to upgrade to something like $ X_{(2)} + X_{(3)} $. I'd like to work this out in general to see if it's balanced, but given that $ X_{(1)}, \dots, X_{(k)} $ are not i.i.d., this has been difficult.