Major advantage of triple spool design is it’s ability to minimize engine surges thus they are the most efficient engine flying.
As mentioned by JETPILOT, namely that the more spools you have, the better because then they will be allowed to spin at their own speed given their spool mass.
Let’s say you’re at takeoff power with for example a double spool engine. The heavy N1 spool is turning at it’s own RPM and so is the lighter N2. Now you chop off the power. The lighter N2 will drop in RPM much quicker then the heavy N1 with it’s huge fan. So what happens is that the N1 compressors are feeding way too much air to the second (HP) compressor.
Where is all that excess air gonna go ?
Well in extreme cases you can have a very damaging engine surge in which air will flow in the wrong way through the engine. Most of todays engines have very sophisticated computer controlled “bypass doors” that let the air escape from the compressor casing.
But the chance of engine surges always remains. If you have three spools, the weight difference is spread over 3 spools thus drastically reducing the chance of engine surges