The correct answer is option 3: Badminton. The ‘Thomas Cup’ is a prestigious international badminton tournament that is held every two years. It was first held in 1949 and is named after Sir George Alan Thomas, an English badminton player and administrator who played a significant role in the development of the sport.

Option 1, Table Tennis, is incorrect as the ‘Thomas Cup’ is not associated with this sport. Table tennis has its own prestigious international tournament called the ‘World Table Tennis Championships’.

Option 2, Lawn Tennis, is also incorrect. The ‘Thomas Cup’ is not related to the sport of lawn tennis. Lawn tennis has its own renowned team competition called the ‘Davis Cup’.

Option 4, Billiards, is not correct either. The ‘Thomas Cup’ is not associated with billiards, which is a different cue sport.

Therefore, the correct answer is option 3: Badminton.