### Step-by-Step Solution:
1. **Understanding the Pauli Exclusion Principle**:
The Pauli Exclusion Principle states that no two electrons in an atom can have the same set of four quantum numbers. This principle is fundamental in quantum mechanics and helps explain the arrangement of electrons in an atom.
2. **Identifying the Four Quantum Numbers**:
The four quantum numbers that define the state of an electron in an atom are:
- **Principal Quantum Number (n)**: Indicates the energy level and size of the orbital.
- **Azimuthal Quantum Number (l)**: Indicates the shape of the orbital (s, p, d, f).
- **Magnetic Quantum Number (m)**: Indicates the orientation of the orbital in space.
- **Spin Quantum Number (s)**: Indicates the spin of the electron, which can be either +1/2 (spin-up) or -1/2 (spin-down).
3. **Application of the Principle**:
According to the principle, while the first three quantum numbers (n, l, m) can be the same for two electrons, their spin quantum number must differ. This means that if two electrons occupy the same orbital (same n, l, and m), they must have opposite spins.
4. **Example**:
For example, consider two electrons in the 4p orbital:
- Electron 1: n = 4, l = 1, m = -1, s = +1/2 (spin-up)
- Electron 2: n = 4, l = 1, m = -1, s = -1/2 (spin-down)
Here, both electrons have the same principal, azimuthal, and magnetic quantum numbers, but different spin quantum numbers, satisfying the Pauli Exclusion Principle.
5. **Conclusion**:
The Pauli Exclusion Principle ensures that no two electrons can occupy the same quantum state within an atom, which is crucial for understanding the structure of the periodic table and the behavior of electrons in atoms.
