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I was taking an online quiz, and I thought option 3 is the correct one, only to find out I got the answer wrong. I know that I could change the answer 2 times and by chance I finally got it right, but I want to understand why option 3 is wrong?
When assuming that the axles of all the rotors are oriented parallel to each other then the thrust will also only be along the same direction as these axles, so this only adds one degree of freedom. The other three degrees of freedom come from the effective torques due to thrust imbalance in front-back (pitch) and left-right (roll), and imbalance in the torque applied by the motors that rotate clockwise-anticlockwise (yaw). The total torque can at most be a three dimensional vector, so adding more rotors with an axle parallel to the others can't increase the span of the possible torque that can be applied.
If you remove the constraint that the rotors all have their axles parallel to each other then six degrees of freedom can be achieved. However most conventional hexacopters do have their axles parallel to each other, so would also only have four degrees of freedom.
The 4 independent inputs directly correspond to the transmitter stick inputs, $u = [\delta_{thr}, \delta_{ele}, \delta_{ail}, \delta_{rud}]^T$
I have no problem with your answer #3, and it could be argued that the question is not sufficiently precise. It depends what you consider the "inputs" to be. At the motor control level, the hex rotor has 6 motors so there are 6 motor speeds that can be independently set. However hexrotors often use the same kind of control handset as a quad rotor, with sticks that control motion:up/down, roll, pitch, yaw. Considered at this level there are 4 "inputs" which are mapped to the 6 motor speeds. This introduces constraints and the 6 motor speeds are no longer independent.
The vehicle itself is a rigid-body moving in 3D space so it has 6 degrees of freedom. Assuming that all propellor axes are parallel, only 4 of those degrees of freedom are directly accessible, just like for a quad rotor. Translational motion is only accessible as a consequence of roll/pitch motion.
