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The set of four quantum numbers is like an address for each electron. Two electrons can't occupy the same space just like how two people can't occupy the same spot, which is why no two electrons can't have the same four quantum numbers in an atom.
To add onto Cassidy's comment above, it's important to know that although each orbital can hold a maximum of 2 electrons, each energy shell can have more than 2 (aside from s) because there are multiple orbitals in each shell.
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