Decision-making authority:
The correct answer is option 'A' - Court. Courts have the authority to make decisions in various matters, including those mentioned in the question. Let's discuss each of these decisions in detail.

1. Decision about a teacher beating a student:
When it comes to a decision about a teacher beating a student, it falls under the purview of the court. The court is responsible for upholding and enforcing laws related to education and child protection. If a student is subjected to physical or mental abuse by a teacher, the court can intervene, investigate the matter, and pass a judgment accordingly. The court ensures that justice is served and the rights of the student are protected.

2. Decision about sharing river waters between states:
Decisions regarding the sharing of river waters between states also come under the jurisdiction of the court. Water disputes between states can be complex and have significant implications for both the states involved and the people relying on those water resources. The court plays a vital role in resolving such disputes by considering various factors like historical usage, equitable distribution, and the overall welfare of all parties concerned. The court's decision ensures fairness and prevents conflicts arising from competing claims over water resources.

3. Decision to punish people for particular crimes:
The decision to punish individuals for specific crimes is also within the authority of the court. The court acts as an impartial arbiter and determines guilt or innocence based on the evidence presented during the trial. If a person is found guilty, the court determines the appropriate punishment, taking into account the severity of the crime, the circumstances, and any mitigating factors. This ensures that justice is served and society is protected from individuals who pose a threat.

In summary, courts have the ultimate authority to make decisions in matters like a teacher beating a student, sharing of river waters between states, and the punishment for specific crimes. The court's role is to ensure justice, protect the rights of individuals, resolve disputes, and maintain law and order in society.