After the death of Shivaji, the founder of the Maratha Empire, there was a battle for succession among his sons and the widow of Shivaji. The correct answer to the question is option 3: Rajaram and Shambhaji.

Shambhaji was the eldest son of Shivaji and he believed that he should be the rightful successor to the throne. On the other hand, Rajaram was the younger son of Shivaji and he also claimed his right to the throne. This led to a power struggle between the two brothers.

Both Shambhaji and Rajaram had their supporters within the empire who sided with them in the fight for succession. Shambhaji had the support of some of the powerful Maratha nobles and military commanders, while Rajaram had the support of his mother, the widow of Shivaji, and some other influential figures.

Ultimately, the battle for succession was won by Rajaram, and he became the Raja of the Maratha Empire. Shambhaji faced a tragic fate as he was captured and executed by the Mughal Emperor Aurangzeb. Rajaram ruled the empire for a brief period before his death, after which his widow Tarabai became a regent for their young son, Shivaji