Final answer:
The maximum number of electrons that can have the quantum numbers n = 3, ℒ = 2, mℓ = 1 is 2, because the two possible spin states (+1/2 or -1/2) allow two distinct configurations for electrons.
Step-by-step explanation:
Considering quantum numbers, the maximum number of electrons that can have the quantum numbers n = 3, ℒ = 2, mℓ = 1 is 2. This is due to the fact that the fourth quantum number, magnetic spin quantum number (ms), which can be either +1/2 or -1/2, allows for two possible states. Electrons having the same n, ℒ, and mℓ must have opposite spins to comply with Pauli's exclusion principle, which states that no two electrons can have the same set of all four quantum numbers (n, ℒ, mℓ, ms).
Since the value of n = 3 sets the primary energy level and ℒ = 2 indicates the d orbital, and with mℓ = 1 specific to one of the five d orbitals' orientation, we simply apply the rule that the spin quantum number ms can be either +1/2 or -1/2, hence, allowing for two electrons in that specific orbital orientation.