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The maximum number of electrons that can occupy a given energy level (or shell) in an atom is determined by the formula (2n^2), where (n) is the principal quantum number corresponding to that energy level. For example, for the first energy level ((n=1)), the maximum is 2 electrons; for the second level ((n=2)), it is 8 electrons; and for the third level ((n=3)), it is 18 electrons. This relationship arises from the quantization of electron states and the constraints imposed by quantum mechanics.

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