Eligible

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late Middle English: via French from late Latin eligibilis, from Latin eligere ‘choose’ (see elect).

wiktionary

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From Middle French eligible, from Latin eligibilis, from ēligō(“select, choose”)

etymonline

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eligible (adj.)

early 15c., "fit or proper to be chosen," from Old French eligible "fit to be chosen" (14c.), from Late Latin eligibilis "that may be chosen," from Latin eligere "choose" (see election). Related: Eligibly.