Option 1: The policy of subsidiary alliance was introduced by Lord Wellesley, not Lord Dalhousie. Under this policy, Indian rulers were required to accept British protection and allow British troops to be stationed in their territories. It did not involve direct annexation.

Option 2: The doctrine of lapse, also known as the policy of annexation, was indeed employed by Lord Dalhousie. According to this doctrine, if a ruler died without a natural heir, the British would not recognize any adopted heir and would annex the state. However, this was not the case with the annexation of Oudh.

Option 3: The correct answer. Oudh was annexed by Lord Dalhousie because it was maladministered and faced severe economic and social problems. The state was accused of misrule and mismanagement, leading to the annexation by the British.

Option 4: There is no historical evidence to suggest that Lord Dalhousie annexed Oudh through waging a war. The annexation was based on the assessment of misrule and maladministration.

In conclusion, option 3, which states that Oudh was annexed due to being maladministered, is the correct answer.