The correct chronological order (beginning with the earliest) of the territories becoming part of India after 15th August, 1947, is option 3: 4-1-2-3.

To break it down:

1. Dadra and Nagar Haveli: It became part of India in 1954. This territory was formerly under Portuguese rule and was incorporated into India through a treaty with Portugal.

2. Goa, Daman, and Diu: These territories were also under Portuguese rule and became part of India in 1961 after the Indian military intervention. Prior to this, Goa, Daman, and Diu were separate territories, but they merged into a single union territory.

3. Pondicherry: It became part of India in 1954. Pondicherry was a French colony, and its union territories, including Pondicherry, Karikal, Mahe, and Yanam, were integrated into India after the French handed over control.

4. Sikkim: It became the 22nd state of India in 1975. Sikkim was a monarchy, and a referendum was conducted, leading to the abolition of the monarchy and the merger of Sikkim with India.

Therefore, the correct chronological order is