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The phrase "the admittance" is correct and usable in written English
It can be used when referring to the act of allowing someone to enter a place or organization. Example: "The admittance of new members to the club will take place next week."
Dictionary
the admittance
noun
The act of admitting.
Exact(58)
But later, she refused to sign the admittance forms.
Often, however, the admittance of new members has engendered controversy.
"My daughter," I say to the nurse at the admittance desk, "she's — they called.
The debate about the admittance of refugees should not, of course, be dominated by economic concerns.
In 1985 the Rabbinical Assembly voted to allow the admittance of women as rabbis for the first time.
Much as I respect Danny Baker, the admittance of brown sauce into his Sausage Sandwich Game is a clear category error.
Enlargement also made integration among EU states harder, a strategic objective of the UK in pushing for the admittance of central and eastern European neighbours.
Newspapers like the Mail and Express, which campaigned against the admittance of Jewish refugees in the 1930s, are now to the fore in attacking Malia.
B represents the reactance of the admittance matrix.
Y redij is the admittance between the generators.
The admittance function Q contains a term of e.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com