Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.

Statements:
All apples are mangos.
All mangos are balls.
No ball is a cat.

Conclusions:
(I): No apple is a cat.
(II): Some balls are apples.

The correct option is: Both conclusions (I) and (II) follow.

Let's analyze the statements step-by-step using logical deductions and set theory (Venn diagrams):

Step 1: Analyze Statement 1 and Statement 2
- "All apples are mangos" means the set of Apples is entirely contained within the set of Mangos (Apples ⊆ Mangos).
- "All mangos are balls" means the set of Mangos is entirely contained within the set of Balls (Mangos ⊆ Balls).
- Combining these two statements, we get: Apples ⊆ Mangos ⊆ Balls. This logically implies that all apples are also balls (Apples ⊆ Balls).

Step 2: Evaluate Conclusion (I): "No apple is a cat"
- We are given Statement 3: "No ball is a cat." This means the set of Balls and the set of Cats have no common elements (Balls ∩ Cats = ∅).
- From Step 1, we know that all apples are balls (Apples ⊆ Balls).
- Since no ball can be a cat, and all apples are inside the set of balls, it is impossible for any apple to be a cat. Therefore, Conclusion (I) logically follows.

Step 3: Evaluate Conclusion (II): "Some balls are apples"
- From Step 1, we established that all apples are balls (Apples ⊆ Balls).
- Since the set of apples is entirely within the set of balls, any element that is an apple must also be a ball. Assuming the existence of apples, there is a portion of the set of balls that contains all the apples.
- Therefore, we can conclude that at least some balls are apples. Thus, Conclusion (II) logically follows.

Conclusion:
Since both Conclusion (I) and Conclusion (II) are logically valid based on the premises, the correct option is "Both conclusions (I) and (II) follow".