The Non-Cooperation Movement, which was a significant part of India`s struggle for independence, started in 1920. This movement was initiated by Mahatma Gandhi as a response to the Jallianwala Bagh massacre and the Rowlatt Act. It aimed to protest against British colonial rule in India through nonviolent means.

Option 1, 1870, is incorrect as it does not align with the historical timeline of the Non-Cooperation Movement. Option 3, 1921, is incorrect as it is the same year the movement gained momentum and spread across the country. Option 4, 1942, is incorrect as it refers to the Quit India Movement.

Therefore, the correct answer is option 2, 1920, as it marks the beginning of the Non-Cooperation Movement.