The other answers are good. Here are some other design considerations:
1) There is a specific design for a certain kind of 2-blade rotor system that is much simpler to manufacture and also cheaper. This is the teetering/underslung hub found on older H-1s and Bell 206s (among others). This type of design has a number of advantages, but one huge disadvantage called mast bumping. This has the effect of limiting the aircraft’s maneuverability in certain situations and pilots have to be careful not to put the aircraft into flight regimes that can cause it. Mast bumping is very bad, often resulting in the separation of the rotor hub from the aircraft. On rotors with 3 or more blades, a teetering hub is impractical and a fully articulated or rigid hub is used (a 2-bladed aircraft could use this). These are more expensive, more complex with more moving parts, but do not suffer from mast bumping.
2) The number of blades affects vibrations. While this might not be a primary design consideration for how many blades a rotor has, an existing aircraft considering adding more blades will possibly need to redesign the vibration absorbing components of the aircraft.
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