Let ___ take ___ book, please. ?

The correct answer to the question "Let ___ take ___ book, please" is option C: "him/you." This choice works well grammatically and contextually in English when constructing a polite request.

In English, "let" is often used with the structure "let + pronoun + verb." Here, we choose "him" to refer to a third party who is being allowed to take the action of taking the book. The pronoun "him" correctly identifies someone else who is not the speaker but is being addressed indirectly.

The second blank asks for the object of the action being taken, which in this case is "the book." By using "you," the speaker is directly addressing the listener, indicating that the listener is involved in the context of the action. The statement can be seen as a polite request for permission or acknowledgment for the person referred to as "him" to take a book that belongs to the person being addressed ("you").

The other options either lack proper grammatical structure or do not effectively convey the intended meaning. Thus, "Let him take your book, please" is the most coherent and contextually fitting choice.