まま

窓(まど)を 開(あ)けた まま で 寝(ね)てしまいました。

The correct answer is "まま". This sentence uses the pattern "V-たまま" (in the state of having V-ed). "開(あ)けた" is the past plain form of the verb "開(あ)ける" (to open). By adding "まま", the phrase "窓(まど)を開(あ)けたまま" describes the state of the window being left open. The particle "で" then connects this state to the following action. "窓(まど)を" is already in the sentence as the object. "開(あ)けた" is the verb in its -ta form, already present. "で" is a particle that follows "まま", not fills the blank before it.

正解は「まま」です。「V-たまま」は「Vした状態のままで」という意味を表します。「開けた」は動詞の「た形」なので「開けたまま」と接続し、「窓を開けた状態のままで」という意味になります。「窓を」と「開けた」はすでに文中にあり、「で」は「まま」の後ろにくる助詞のため不適当です。