Option 1: Auckland Colvin is not the correct answer. He was a British civil servant who served as the Lieutenant Governor of the North-Western Provinces and Oudh in India during the 19th century. He was not involved in the founding of the Mohammedan Anglo-Oriental Defence Association.

Option 2: Badruddin Tyabji is not the correct answer. He was an Indian lawyer and activist who served as the third President of the Indian National Congress in 1887. Although he was an influential figure in the Indian independence movement, he did not found the Mohammedan Anglo-Oriental Defence Association.

Option 3: Theodore Beck is not the correct answer. There is limited information available on Theodore Beck and his connection to the Mohammedan Anglo-Oriental Defence Association. Therefore, he is unlikely to be the correct answer.

Option 4: Sir Syed Ahmad Khan is the correct answer. He was a prominent Muslim philosopher, social reformer, and politician in British India. He founded the Mohammedan Anglo-Oriental Defence Association (later renamed as All India Muslim League) in 1893. The association aimed to promote and protect the interests of the Muslim community in India.

In conclusion, the correct answer is option 4