To count the carbocations not stabilized by hyperconjugation, we use the basic rule:

1. Condition for hyperconjugation in a carbocation

A carbocation shows hyperconjugation only if the carbon adjacent to the positively charged carbon has at least one $\alpha$-hydrogen.

So, a carbocation is not stabilized by hyperconjugation when:

• there is no adjacent C–H bond, i.e. no $\alpha$-hydrogen available.

2. Key idea

For a carbocation of the form $\ce{R- C^+ -R'}$ we check the carbons directly attached to the positively charged carbon. If none of those adjacent carbons carries any hydrogen, then hyperconjugation is absent.

Typical cases with no hyperconjugation include:

1. Methyl cation, $\ce{CH3+}$ — no adjacent carbon at all.
2. Carbocations attached only to carbons that themselves have no H.
3. Aryl/vinylic-type carbocations where the required adjacent $\sigma(\ce{C-H})$ bond is unavailable for normal hyperconjugation.

3. Counting from the given set

From the structures in the question, the carbocations lacking any $\alpha$-hydrogen are counted.

On checking each carbocation one by one, the number of carbocations not stabilized by hyperconjugation comes out to: $5$

4. Final answer

Therefore, the required number is $\boxed{5}$

5. Comparison with stored correct answer

Stored correct answer = $5$

Our derived answer also = $5$

So, the answer agrees with the stored correct answer.