Match the entries of column I with appropriate entries of column II and choose the correct option out of the four options given.

(i)-(c), (ii)-(a), (iii)-(d), (iv)-(b)

The correct answer is option 2. (i)-(c), (ii)-(a), (iii)-(d), (iv)-(b).

Let us delve into the details of the oxidation states of manganese \((Mn)\) and lanthanoids (lanthanides), and how they match with the entries in Column I and Column II.

(i) Oxidation state of \(Mn\) in \(MnO_2\): (c) +4:

Manganese dioxide \((MnO_2)\) is a common compound where manganese exhibits the \(+4\) oxidation state. The formula \(MnO_2\) indicates that manganese is bonded to two oxygen atoms, and the oxidation state of oxygen is typically \(-2\). Therefore, to balance the charges, manganese must be in the \(+4\) oxidation state.

(ii) Most stable oxidation state of Mn:(a) +2: 

The most stable oxidation state of manganese is \(+2\). In compounds like manganese(II) chloride \((MnCl_2)\) or manganese(II) sulfate \((MnSO_4)\), manganese is in the \(+2\) oxidation state.

(iii) Most stable oxidation state of \(Mn\) in oxides: (d) +7:

Manganese in oxides typically exhibits a variety of oxidation states depending on the compound. However, manganese dioxide \((MnO_2)\) is a common oxide where manganese is in the \(+4\) oxidation state.

(iv) Characteristic oxidation state of lanthanoids: (b) +3: 

Lanthanoids (lanthanides) are a group of elements in the periodic table with atomic numbers 57 to 71. They typically exhibit the \(+3\) oxidation state as their characteristic oxidation state. Compounds such as lanthanum(III) oxide \((La_2O_3)\) or cerium(III) chloride \((CeCl_3)\) illustrate the +3 oxidation state of lanthanoids.

Upon careful analysis of the oxidation states of manganese and lanthanoids, the correct matching of entries from Column I with those in Column II is: (i)-(c), (ii)-(a), (iii)-(c), (iv)-(b).

This alignment reflects the typical oxidation states observed for manganese and lanthanoids in various compounds and contexts as discussed.