In this question, three statements are given, followed by three conclusions numbered I, II and III. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusion(s) logically follows/follow from the statements.

Statements:

Some carrots are onions.

All onions are potatoes.

All potatoes are radish.

Conclusions:

I. Some carrots are radish.

II. All onions are radish.

III. No carrot is a potato. 

The least possible Venn diagram for the given statements is as shown below :

Conclusions:

I. Some carrots are radish → Follow (As some carrots are onions, all onions are potatoes and all potatoes are radish. As some part of carrots is common with onions which whole comes in potatoes which whole comes in radish, therefore some carrots are radish is true.)

II. All onions are radish → Follow (As all onions are potatoes and all potatoes are radish. As some part of carrots is common with onions which whole comes in potatoes which whole comes in radish, therefore all onions are radish is true.)

III. No carrot is a potato → Does not follow (As some carrots are onions and all onions are potatoes. As no direct relation is given between carrot and potato, therefore no carrot is potato is false.)

∴ Here, Both conclusions I and II follow.​

Hence, the correct answer is "Both conclusions I and II follow"