Consider something like OLS with edges already oriented, still looking at only the FR slot. Note that this time, we don't have the FR pair pre-formed. We start with the DFR corner in any of five possible locations (momentarily ignoring its orientation); the FR edge in any of five possible locations; the LSLL corners in any of 3^(5−1) = 81 possible orientations; and we want to solve all of these. Multiply these together, and you get 2025 cases in total.

But this time you also have to determine whether you care about pre-AUF! (Post-AUF doesn't change the solved state, so we don't need to care.) Do you consider

R U R' and

U2 R U R' to be algs for the same case? Let's say you do. Here, we can split the cases into four subsets based on where the FR edge and DFR corner are.

(a) If they're both in the slot, you're free to AUF the top layer however, which turns out to have 8 cases for the top layer and 3 cases for how the DFR corner is twisted, so 24 cases in total (1 of which is solved and 7 of which are normal OLL cases).

(b) If only the FR edge is in the slot, you can AUF the DFR corner to URF, so we can repeat the above computation with the necessary adjustments to get 3^(5−1) = 81 cases.

(c) Likewise, if only the DFR corner is in the slot, we also get 81 cases.

(d) The last bunch of cases is where both pieces aren't in the slot; we can AUF the DFR corner to URF, and we get 4 (location of FR) × 3^(5−1) = 324 cases in this subset.

Add these all up, and we have 510 cases in total for OLS with oriented edges.