Explanation:

The stability of a carbocation depends on the electron donating or withdrawing ability of the substituent groups attached to it. In this case, we have to compare the stability of four carbocations, which are:

a) C6H5CH2+
b) C6H5CH2CH2+
c) C6H5CH+CH3
d) C6H5C+(CH3)2

To compare their stability, we have to look at the following factors:

1. Inductive effect: It is the electron-withdrawing or donating effect of the substituent groups attached to the carbocation. The more electron-donating the groups, the more stable the carbocation.

2. Hyperconjugation effect: It is the stabilizing effect of the adjacent C-H bonds that donate electrons to the empty p-orbital of the carbocation.

3. Resonance effect: If the carbocation can form a resonance structure, it will be more stable.

Now, let's analyze each carbocation and compare their stability:

a) C6H5CH2+: It has a phenyl (C6H5) group, which is electron-donating. However, it has no adjacent C-H bonds to donate electrons to the empty p-orbital. Also, it cannot form a resonance structure. Therefore, it is the least stable carbocation among the given options.

b) C6H5CH2CH2+: It has two ethyl (CH2CH3) groups, which are weakly electron-donating. It has four adjacent C-H bonds to donate electrons to the empty p-orbital. However, it cannot form a resonance structure. Therefore, it is more stable than option a), but less stable than options c) and d).

c) C6H5CH+CH3: It has a methyl (CH3) group, which is weakly electron-donating. It has three adjacent C-H bonds to donate electrons to the empty p-orbital. Also, it can form a resonance structure, where the positive charge can be delocalized over the phenyl ring. Therefore, it is more stable than options a) and b), but less stable than option d).

d) C6H5C+(CH3)2: It has two methyl (CH3) groups, which are strongly electron-donating. It has six adjacent C-H bonds to donate electrons to the empty p-orbital. Also, it can form a resonance structure, where the positive charge can be delocalized over the phenyl ring and both methyl groups. Therefore, it is the most stable carbocation among the given options.

Therefore, the correct answer is option d) C6H5C+(CH3)2.