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Instead of writing out the entire electron configuration starting at 1s for an element, using the brackets of the closest previous noble gas is a shortcut and the configuration for an element will continue from there.
We do this as an easy way to shorten the written electron configuration. Inside of the brackets is the closest noble gas from the row above, since their electron configuration doesn't change, and then next to it we write out the next electrons that our element has above the noble gas.
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