Given below are two statements:

Statement I: $$(CH_3)_3C^+$$ is more stable than $$CH_3^+$$ as nine hyperconjugation interactions are possible in $$(CH_3)_3C^+$$.
Statement II: $$CH_3^+$$ is less stable than $$(CH_3)_3C^+$$ as only three hyperconjugation interactions are possible in $$CH_3^+$$.

In the light of the above statements, choose the correct answer from the options given below

Hyperconjugation is a stabilizing effect in carbocations where electrons from the C-H bond adjacent to the carbocation carbon (alpha hydrogens) are donated into the empty p-orbital of the carbocation. The number of hyperconjugation interactions depends on the number of alpha hydrogens.

Consider the methyl carbocation, $$CH_3^+$$. The carbocation carbon has no alkyl groups attached, so there are no alpha carbons and hence no alpha hydrogens. Therefore, the number of hyperconjugation interactions in $$CH_3^+$$ is zero.

Now, consider the tert-butyl carbocation, $$(CH_3)_3C^+$$. The carbocation carbon is attached to three methyl groups. Each methyl group has three hydrogens, so the total number of alpha hydrogens is 3 × 3 = 9. Therefore, nine hyperconjugation interactions are possible.

Statement I claims that $$(CH_3)_3C^+$$ is more stable than $$CH_3^+$$ because it has nine hyperconjugation interactions. This is correct because more hyperconjugation interactions lead to greater stability. Thus, Statement I is true.

Statement II claims that $$CH_3^+$$ is less stable than $$(CH_3)_3C^+$$ because it has only three hyperconjugation interactions. However, as established, $$CH_3^+$$ has zero hyperconjugation interactions, not three. Therefore, Statement II is false.

In summary, Statement I is true and Statement II is false.

The correct option is C: Statement I is true but Statement II is false.