To correctly match List I with List II concerning medical conditions and their associated chromosomes, it's essential to have some understanding of genetics and chromosome abnormalities related with each condition. Here's the correct matching based on that information:

• Down's syndrome: This genetic disorder is characterized by an extra copy (trisomy) of the 21$^\mathrm{st}$ chromosome, which makes it associated with 21$^\mathrm{st}$ chromosome (III).
• $\alpha$-thalassemia: This condition is related to a mutation or deletion in the alpha globin genes located on the 16$^\mathrm{th}$ chromosome (IV).
• $\beta$-thalassemia: This is a blood disorder caused due to mutations in the beta globin gene found on the 11$^\mathrm{th}$ chromosome (I).
• Klinefelter's syndrome: This syndrome arises from the presence of an extra X chromosome in males (usually XXY), so it is related to abnormality in the 'X' chromosome (II).

Given these matches:

• A - III
• B - IV
• C - I
• D - II

The correct option based on the details above is:

Option C: A-III, B-IV, C-I, D-II