1. Identify the radicals

The given radicals are:

• ${(CH_3)_2\dot{C}H}$ : secondary alkyl radical
• ${(CH_3)_3\dot{C}}$ : tertiary alkyl radical
• ${((C_6H_5)_2\dot{C}H)}$ : diphenylmethyl radical
• ${((C_6H_5)_3\dot{C})}$ : triphenylmethyl radical

2. Recall the factors affecting free radical stability

Free radicals are stabilized mainly by:

• Hyperconjugation from alkyl groups
• Resonance delocalization into adjacent $\pi$ systems

Now compare:

• Alkyl radicals are stabilized by hyperconjugation.
• Benzylic/phenyl-substituted radicals are much more stabilized because the unpaired electron can delocalize by resonance over phenyl rings.
• More phenyl rings attached to the radical center $\Rightarrow$ more resonance stabilization.

Thus,

• ${(CH_3)_3\dot{C}}$ is more stable than ${(CH_3)_2\dot{C}H}$ due to greater hyperconjugation.
• ${((C_6H_5)_2\dot{C}H)}$ is more stable than ${(CH_3)_3\dot{C}}$ because resonance stabilization from two phenyl rings is stronger than alkyl hyperconjugation.
• ${((C_6H_5)_3\dot{C})}$ is the most stable because the unpaired electron is delocalized over three phenyl rings.

3. Write the increasing order

Therefore, the increasing order of stability is:

4. Match with the options

This corresponds to Option B.

5. Comparison with stored answer

Stored correct answer = B

Our derived answer = B

So, the answers agree.