[{_id:"some_id",details:[{degree:{value:"ABC"},iseligible:{value:"YES"}}]},{_id:"this_is_an_id",details:[{degree:{value:"XYZ"},iseligible:{value:"YES"}}]},{_id:"some_other_id",details:[{degree:{value:"DEF"},iseligible:{value:"NO"}}]},{_id:"another_id",details:[{degree:{value:"GHI"},iseligible:{value:"NO"}}]}]

db.collection.aggregate([{$match:{"details.iseligible.value":"YES"}},{$count:"numDocsFound"}])