I think you want to do the second option, it would be easier. Just start with all four in parallel and change one at a time to end up with four in series.

Well.. The best way that i still think is the best is to draw out all 10 cicuits.. However, you can still try to do this using some maths.

Let's not make this too complicated.. Split up the types of arrangements first.
Let resisitors in parallel be considered as 1 resistor.
Therefore, we have: ....4R....3R....2R....1R... .....|......|......|......|... .....1......1......1......1...

For 4 resistors in parallel, we have only 1 type of arrangement(4 R in Parallel)

For 3 resistors in parallel, we will have 2 types of arrangement (1 R in series with the other 3 R in parallel)(Rotate bout the positions of the resistors in series to get 2)/ You can also permute 2 with 2..

For 2 R in parallel, we will have 6 types of arrangement (2 R in series with the other 2 in parallel) Taking the 2 parallel R as 1 R, Permute the 3 R with the 3 slots. ie: 3P3=6

For all resistors in series, we will have only one type of arrangement.