The quickest way to settle this is to ask a single question: which Part of the Constitution does the provision live in? Part III holds fundamental rights and courts enforce them. Part IV holds directive principles and Article 37 says no court can enforce those. Part IVA holds fundamental duties, which bind citizens rather than the State.
The right to free and compulsory education for children aged six to fourteen is Article 21A, and Article 21A sits in Part III. The Constitution (Eighty-sixth Amendment) Act, 2002 put it there. A child denied a school place can move the High Court under Article 226 or the Supreme Court under Article 32. That makes it a fundamental right enforceable in law.
The history is worth carrying. In Unni Krishnan v. State of Andhra Pradesh the Supreme Court drew a right to education for children up to fourteen out of the right to life in Article 21. The Eighty-sixth Amendment then wrote it in expressly, and the 2009 Act supplied the working details.
Let's summarize:
So option (A) is the answer.