In modern Indian history, by whom was the Home Rule Movement in India formally started as a campaign for self government within the British Empire?

Introduction / Context:
The Home Rule Movement is a key topic in the Indian freedom struggle. It aimed to secure self government for India within the British Empire and was inspired partly by similar movements in other parts of the world. Knowing who led and organised this movement helps you understand the transition from moderate constitutional methods to mass mobilisation in the early twentieth century.

Given Data / Assumptions:
- The question asks who started the Home Rule Movement in India.
- Options pair different leaders such as Annie Besant, Bal Gangadhar Tilak, Mahatma Gandhi and others.
- We assume the focus is on the two main Home Rule Leagues formed during this movement.

Concept / Approach:
The movement featured two Home Rule Leagues, one led by Bal Gangadhar Tilak and another by Annie Besant. Both leaders campaigned vigorously for self rule and used meetings, pamphlets and tours to spread the idea. Recognising that these two names are always associated with Home Rule simplifies the question significantly, as other leader combinations listed in the options are associated with different phases and movements.

Step-by-Step Solution:
Step 1: Recall that Annie Besant founded the Home Rule League in Madras and later expanded it to other parts of India. Step 2: Remember that Bal Gangadhar Tilak started a separate Home Rule League based in Maharashtra and surrounding regions. Step 3: Note that both leaders shared the common goal of demanding self government for India within the British Empire. Step 4: Identify the option that includes both Annie Besant and Bal Gangadhar Tilak together. Step 5: Confirm that none of the other options reflect the recognised leadership of the Home Rule Movement.

Verification / Alternative check:
To verify, connect the Home Rule Movement to the period just before Mahatma Gandhi became the central figure in the national movement. At that time, Tilak had returned to active politics after his imprisonment, and Annie Besant was an influential political and social leader. Textbooks and exam guides consistently mention them as the two leaders who started and led the Home Rule Leagues. This pattern confirms the correct choice.

Why Other Options Are Wrong:
S N Banerjee and W C Banerjee were important early nationalist leaders but were not the main organisers of the Home Rule Movement.
Mahatma Gandhi and Motilal Nehru worked together in later phases of the freedom struggle, especially during mass civil disobedience, not in starting the Home Rule Movement.
Annie Besant and Gopal Krishna Gokhale were both moderates at different times, but the movement was not jointly started by them.
Subhash Chandra Bose and Jawaharlal Nehru belong to a later radical phase involving the Indian National Army and political leadership on the eve of independence, not the Home Rule period.

Common Pitfalls:
Some examinees may incorrectly include Mahatma Gandhi in every major movement of the freedom struggle, ignoring the specific chronology. Others might confuse Annie Besant and Gopal Krishna Gokhale because both are associated with early moderates. To avoid these mistakes, organise modern Indian history in a timeline, clearly placing the Home Rule Movement between the early moderates and the era of Gandhian mass movements.

Final Answer:
The Home Rule Movement in India was started and led by Annie Besant and Bal Gangadhar Tilak through their respective Home Rule Leagues.