Out of four cities given below three are alike in some manner while the fourth…

Concept: In a classification (odd-one-out) item, the members are tested against one shared property at a time. The intended property is the one that splits the group 3-versus-1 — exactly three members possess it and one does not. A property that splits the group 2-versus-2, or 2-1-1, is not the intended basis, because it fails to isolate a single member.

Application: test the candidate properties against these four cities, one at a time.

1. State: Lucknow and Allahabad lie in Uttar Pradesh, Rishikesh lies in Uttarakhand and Patna lies in Bihar. This is a 2-1-1 split, so no single city is isolated.

2. State capital: Lucknow is the capital of Uttar Pradesh and Patna is the capital of Bihar, while Rishikesh and Allahabad are not capitals. This is a 2-versus-2 split, so again no single city is isolated.

3. River on whose bank the city stands: Rishikesh, Allahabad and Patna all stand on the Ganga, while Lucknow stands on the Gomti. This is a clean 3-versus-1 split.

Cross-check: compare all four cities on every property at once.

Result: only the river test isolates exactly one city. Lucknow is therefore the odd one out, because it is the only one of the four that does not stand on the Ganga; it stands on the Gomti. Allahabad, renamed Prayagraj, still counts as a Ganga city, since the Sangam is the point where the Yamuna joins the Ganga.

Note: a classification item can admit more than one grouping property — city size, for instance, separates the smaller town from the three million-plus cities. The basis accepted in the official answer key for this question, and the property these four cities are most commonly grouped by, is the river on which each stands.