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The correct answer is option 2: Meghalaya. Meghalaya does not share a border with Bhutan.
Option 1: Sikkim shares a border with Bhutan. It is located in the northeastern part of India and shares its southern border with Bhutan.
Option 2: Meghalaya does not share a border with Bhutan. Meghalaya is a state located in northeastern India, and it shares its borders with Assam and Bangladesh, but not Bhutan.
Option 3: West Bengal shares a border with Bhutan. It is located in eastern India, and its northern part shares a border with Bhutan.
Option 4: Arunachal Pradesh shares a border with Bhutan. It is located in northeastern India and shares its western border with Bhutan.
Therefore, the correct answer is option 2: Meghalaya. It does not share a border with Bhutan.