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If nitrogen has 7 electrons, how come the ground state is $2s^2 2p^3$? This would mean that there are only 5 electrons in the nitrogen atom.

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The full electronic configuration of nitrogen is $1s^2 \, 2s^2 \, 2p^3$, and the expression $2s^2 \, 2p^3$ is an abbreviated form.

This is a general case of a more general usage: if an atom has a completely filled inner shell (such as $1s^2$, $2s^2 \, 2p^6$, $3s^2 \, 3p^6$, $3d^{10} \, 4s^2 \, 4p^6$, and so on), then the inner shells can be omitted, and they can be replaced with the corresponding noble-gas atom. Thus, for nitrogen, you could also express the configuration as $[\mathrm{He}]\,2s^2 \, 2p^3$ (which is how Wikipedia reports it).

The reason is that inner shells are rarely relevant, and repeating them all over the place makes for bulky and inconvenient notation that is not conveying any useful information. Notation is there to help convey ideas: it needs to be unambiguous and clear, but beyond that, there is a big premium on conciseness and economy, which is why non-essential information (which can be inferred from the context anyway) isn't very useful to keep around.

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It actually is 1s$^2$2s$^2$2p$^3$.You can find this by searching "nitrogen ground state configuration".

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