The Electron-Ion Collider (EIC) at Brookhaven National Laboratory will be a precision Quantum Chromodynamics machine that will enable a vast physics program with electron+proton/ion collisions across a broad center-of-mass range. Measurements of hard probes such as heavy flavor in deep inelastic scatterings will be an essential component to the EIC physics program and are one of the detector R&D driving aspects. In this paper we study the projected statistical precision of open charm hadron production through exclusive hadronic channel reconstruction with a silicon detector concept currently being developed using a PYTHIA-based simulation. We further study the impact of possible intrinsic charm in the proton on projected data, and estimate the constraint on the nuclear gluon parton distribution function (PDF) from the charm structure functions $F_{2}^{coverline{c}}$ in $e$+Au collisions using a Bayesian PDF re-weighting technique. Our studies show the EIC will be capable delivering an unprecedented measurement of charm hadron production across a broad kinematic region and will provide strong constraints to both intrinsic charm and nuclear gluon PDFs.
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