non-admission n.

a prefix used freely in English and meaning "not, lack of," or "sham," giving a negative sense to any word, 14c., from Anglo-French noun-, from Old French non-, from Latin non "not, by no means, not at all, not a," from Old Latin noenum "not one" (*ne oinom, from PIE root *ne- "not" + PIE root *oi-no- "one, unique"). In some cases perhaps from Middle English non "not" (adj.), from Old English nan (see not). "It differs from un- in that it denotes mere negation or absence of the thing or quality, while un- often denotes the opposite of the thing or quality" [Century Dictionary].

early 15c., "acceptance, reception, approval," from Latin admissionem (nominative admissio) "a letting in," noun of action from past-participle stem of admittere "admit, give entrance; grant an audience," of acts, "let be done, allow, permit," from ad "to" (see ad-) + mittere "let go, send" (see mission).

The meaning "an acknowledging" is from 1530s. The literal sense of "act of allowing to enter, admittance," is from 1620s. As short for admission price, by 1792.