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An electron cannot have the quantum numbers ( n=1, \ell=1, m_\ell=0, m_s=-\frac{1}{2} ) because the principal quantum number ( n ) must be a positive integer and the azimuthal quantum number ( \ell ) must satisfy ( 0 \leq \ell < n ). Since ( n=1 ) allows only ( \ell=0 ), the specified ( \ell=1 ) is not permissible. Therefore, the set of quantum numbers violates the rules of quantum mechanics, making it impossible for an electron to possess them.
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