This question involves understanding carbocation stability and hydride shift rearrangements. A hydride shift (H⁻ shift) occurs when a hydrogen atom with its bonding electrons moves from one carbon to an adjacent carbocation to form a more stable carbocation. The rearrangement from B → A will occur if the carbocation A is more stable than carbocation B.

Key stability order for carbocations:
Tertiary (3°) > Secondary (2°) > Primary (1°) > Methyl

Step 1: Analyze Each Pair

First Pair:
A: Me₂CHCH₂⁺ (isobutyl cation) – This is a primary carbocation.
B: Me₃C⁺ (tert-butyl cation) – This is a tertiary carbocation.
Since tertiary is more stable than primary, the rearrangement B → A will NOT occur (it would actually be A → B).

Second Pair:
A: MeCH₂CH₂⁺ (n-propyl cation) – This is a primary carbocation.
B: Me₂CH⁺ (isopropyl cation) – This is a secondary carbocation.
Since secondary is more stable than primary, the rearrangement B → A will NOT occur (it would be A → B).

Third Pair:
Let's write the structures clearly. The formatting in the question is slightly ambiguous but can be interpreted.
A: MeCH₂CH₂C⁺HCH₃ – This is ${\mathrm{CH}}_3{\mathrm{CH}}_2{\mathrm{CH}}_2\stackrel{+}{C}H{\mathrm{CH}}_3$. The carbocation carbon is bonded to one carbon and two hydrogens, making it a primary carbocation.
B: MeCH₂CH₂CH₂C⁺H₂ – This is ${\mathrm{CH}}_3{\mathrm{CH}}_2{\mathrm{CH}}_2{\mathrm{CH}}_2\stackrel{+}{C}{H}_2$. The carbocation carbon is bonded to no other carbons (only to two H's), making it a methyl carbocation (even less stable than primary).
A primary carbocation is more stable than a methyl carbocation. Therefore, the rearrangement B → A WILL occur to form the more stable ion.

Final Answer: The rearrangement B → A takes place only in the 3rd Pair.

Related Topics & Formulae

Carbocation Stability: The stability of carbocations increases with the number of alkyl groups attached to the positively charged carbon due to the electron-donating inductive effect and hyperconjugation. The order is 3° > 2° > 1° > CH₃⁺.

Rearrangements: Carbocations can undergo rearrangements (like hydride or alkyl shifts) to form more stable carbocations. This is a key concept in reaction mechanisms, especially in SN1 reactions and electrophilic additions to alkenes.