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(C) Hyperconjugation requires the presence of at least one $\alpha$-hydrogen atom on a carbon atom adjacent to the positively charged carbon atom.Let

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$3$

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(C) Hyperconjugation requires the presence of at least one $\alpha$-hydrogen atom on a carbon atom adjacent to the positively charged carbon atom.Let us analyze the given carbocations:$1$. $sec$-butyl carbocation: $CH_3-CH^+-CH_2-CH_3$ has $5$ $\alpha$-hydrogens (stabilized).$2$. $Di(tert-butyl)$ methyl carbocation: $(t-Bu)_2CH^+$ has $2$ $\alpha$-hydrogens (stabilized).$3$. Methyl carbocation: $CH_3^+$ has no $\alpha$-hydrogen (not stabilized).$4$. Cyclopentadienyl cation: This is an anti-aromatic system,and the positive charge is on an $sp^2$ carbon with no $\alpha$-hydrogen (not stabilized).$5$. Methoxymethyl cation: $CH_3-O-CH_2^+$ is stabilized by resonance from the lone pair on oxygen,but it has no $\alpha$-hydrogen (not stabilized).$6$. Isopropyl carbocation: $(CH_3)_2CH^+$ has $6$ $\alpha$-hydrogens (stabilized).$7$. Dimethylaminomethyl cation: $(CH_3)_2N-CH_2^+$ is stabilized by resonance from the lone pair on nitrogen,but it has no $\alpha$-hydrogen (not stabilized).Thus,the carbocations not stabilized by hyperconjugation are: Methyl carbocation,Cyclopentadienyl cation,Methoxymethyl cation,and Dimethylaminomethyl cation.However,re-evaluating the provided image list: $sec$-butyl,$di(tert-butyl)methyl$,$CH_3^+$,cyclopentadienyl,$CH_3OCH_2^+$,isopropyl,and $(CH_3)_2NCH_2^+$.Total count of carbocations not stabilized by hyperconjugation is $4$.

Number of carbocations from the following that are not stabilized by hyperconjugation is $...........$ .

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