“Admission” is a much more common word than “admittance” and is a good choice for almost all contexts. You may gain admission or admittance to a college, but you’ll probably be dealing with its admissions office. When “admittance” is used, it’s most likely to refer to physical entry into some place or other, as is indicated by signs saying “No Admittance.”
In electronics, admittance is the opposite of impedance.
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