**Explanation:**

The [4 + 2] cycloaddition reaction between 1,3-butadiene and ethylene involves the overlap of the highest occupied molecular orbital (HOMO) of one reactant with the lowest unoccupied molecular orbital (LUMO) of the other reactant. Let's analyze the overlap of the HOMO and LUMO orbitals in this reaction:

**a) Overlap of the HOMO of butadiene with the LUMO of ethylene:**

- The HOMO of butadiene is a π molecular orbital, formed by the overlap of the p orbitals of the carbon atoms in the conjugated system. It consists of two nodal planes, with electron density both above and below the π system.
- The LUMO of ethylene is also a π molecular orbital, formed by the overlap of the p orbitals of the carbon atoms in the double bond. It consists of a single nodal plane in the middle of the π system, with electron density on either side of the plane.
- The overlap of the HOMO of butadiene with the LUMO of ethylene occurs when the π system of butadiene approaches the double bond of ethylene.
- The electron density in the HOMO of butadiene interacts with the electron-deficient region of the LUMO of ethylene, resulting in the formation of a new σ bond between the two reactants.
- This overlap leads to the formation of a cyclohexene derivative as the product of the [4 + 2] cycloaddition reaction.

**b) Overlap of the HOMO of ethylene with the LUMO of butadiene:**

- The HOMO of ethylene is also a π molecular orbital, formed by the overlap of the p orbitals of the carbon atoms in the double bond. It consists of a single nodal plane in the middle of the π system, with electron density on either side of the plane.
- The LUMO of butadiene is a π* molecular orbital, formed by the overlap of the p orbitals of the carbon atoms in the conjugated system. It also consists of two nodal planes, with electron density both above and below the π system.
- The overlap of the HOMO of ethylene with the LUMO of butadiene occurs when the double bond of ethylene approaches the π system of butadiene.
- However, this overlap does not lead to the formation of a new bond or the [4 + 2] cycloaddition product. Instead, it leads to a non-productive interaction between the two reactants.

**c) Both (a) and (b):**

- As discussed above, the overlap of the HOMO of butadiene with the LUMO of ethylene results in the formation of a new σ bond and the [4 + 2] cycloaddition product.
- On the other hand, the overlap of the HOMO of ethylene with the LUMO of butadiene does not lead to the formation of a new bond or the desired cycloaddition product.
- Therefore, the correct answer is option 'c' - both (a) and (b) - as the productive overlap occurs between the HOMO of butadiene and the LUMO of ethylene.