I don't think there are only four solutions. What about, for example, ( N, T, V, O, S )?
The rules say that each piece shares one instrument with the piece performed immediately before or after it (or both).
So, ( N, T, V, O, S ) would have the instruments (FL, FG, LM, HM, HG). Though the second and third pieces performed do not share an instrument in common, the third and fourth, along with the fourth and fifth, share an instrument. There are many more than four possibilities, though the four listed serve as a good starting point to tackle the game.