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Option 1 : Only II

In the given question,

Statement Correctness Reason
At least one girl has blue eyes Incorrect This could be a probable situation but not sure as maybe both boys have blue eyes and both girls have brown eyes.
Two of the children are boys. Correct This is a sure situation as among 4 children half are girls that mean other half being 2 are boys.
The boys have brown eyes. Incorrect Probable but not sure as maybe both girls has brown eyes and both boys have blue eyes, or one girl and one boy has brown eyes.

 

Therefore, Option 1 is the correct answer.

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