Hyperconjugation

Hyperconjugation stabilises a carbocation through overlap of the sigma bonding orbital of an adjacent C-H bond with the empty p orbital on the positive carbon, and the number of such contributing structures equals the number of alpha hydrogens. In the tertiary butyl cation, (CH_3)_3C^+, the central positive carbon is bonded to three methyl groups, and each methyl carbon carries three hydrogens that are alpha to the cationic centre. Therefore the total count of alpha hydrogens is three groups times three hydrogens, equal to nine, giving nine hyperconjugative structures. The answer 6 wrongly counts only two methyl groups. The answer 3 counts the hydrogens of a single methyl group, ignoring the other two. The answer 1 underestimates drastically and contradicts the symmetric structure. This maximum delocalisation explains why tertiary cations are so stable, consistent with NCERT, and the same alpha-hydrogen count governs the relative stability of alkenes and free radicals as well. Each hyperconjugative structure is a no-bond resonance form in which a C-H bond is shown as broken and its electrons shared with the empty orbital, so more alpha hydrogens mean more such forms and greater stabilisation. Plausibility check: a more stable cation should have more alpha hydrogens, and nine is the largest count among simple alkyl cations, supporting the high stability of the tertiary butyl ion.