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The correct answer is: I and III only.
The Chairman of the Finance Commission is appointed by the President of India. The Finance Commission is a constitutional body responsible for recommending the distribution of financial resources between the central government and the state governments in India.
The Chief Minister of a Union Territory is also appointed by the President of India. Union Territories in India are administered by a Lieutenant Governor appointed by the President, and the Chief Minister, if applicable, is appointed by the Lieutenant Governor.
However, it is important to note that the Planning Commission, mentioned in option II, has been replaced by the NITI Aayog (National Institution for Transforming India), and the Deputy Chairman of the Planning Commission is not appointed by the President anymore. Therefore, option II is not correct.
So, the correct answer is: I and III only.