The river that originates at Amarkantak is the Narmada, making option 3 the correct answer.

Amarkantak is a significant pilgrimage site located in the Indian state of Madhya Pradesh. It is known as the "King of Pilgrimage Sites" because it is the source of several rivers. Among these rivers, the Narmada holds great importance.

The Narmada River is one of the longest rivers in India, stretching over 1,300 kilometers in length. It flows through the central part of the country, primarily through the states of Madhya Pradesh, Maharashtra, and Gujarat. The river eventually drains into the Arabian Sea.

Amarkantak is situated at an altitude of approximately 1,048 meters, where the Vindhya and Satpura mountain ranges converge. It is from this sacred place that the Narmada River originates. The river emerges as a small spring called the Narmada Kund, located within the Amarkantak plateau.

The Narmada River is regarded as one of the seven holy rivers in Hinduism. It has a rich cultural and historical significance in Indian mythology and is often associated with devotion and spirituality. The river has been revered by pilgrims and worshipped as a goddess by many communities residing along its banks.

Moreover, the Narmada River has played a vital role in the socio-economic development of the regions it traverses. It supports irrigation for agricultural activities, serves as a source of drinking water, and facilitates hydroelectric power generation through various dams constructed along its course.

In contrast, let`s briefly discuss the other options:

1. Damodar: The Damodar River is a river in eastern India, primarily flowing through the states of Jharkhand and West Bengal. It originates from the Chota Nagpur Plateau and is known as the "Sorrow of Bengal" due to its history of devastating floods. It does not originate at Amarkantak.

2. Mahanadi: The Mahanadi River is a major river in East Central India, flowing through the states of Chhattisgarh and Odisha. It originates in the highlands of the Raipur district of Chhattisgarh. It also does not originate at Amarkantak.

4. Tapti: The Tapti River, also known as the Tapi River, is a river in western India, primarily flowing through the states of Maharashtra and Gujarat. It originates in the Betul district of Madhya Pradesh. Similar to the previous options, it does not have its source at Amarkantak.

In conclusion, the river that originates at Amarkantak is the Narmada River, making option 3 the correct answer.