1. Identify all possible positional isomers of dimethylcyclopentane

For a cyclopentane ring with two identical methyl groups, the distinct positional isomers are:

• $1,1$-dimethylcyclopentane
• $1,2$-dimethylcyclopentane
• $1,3$-dimethylcyclopentane

(Note: $1,4$ is equivalent to $1,3$ in cyclopentane by renumbering.)

2. Count stereoisomers for each positional isomer

(i) $1,1$-Dimethylcyclopentane

Both methyl groups are on the same carbon. That carbon does not become a stereocenter because it has two identical substituents ($\ce{CH3}$ and $\ce{CH3}$).

So, number of stereoisomers: $1$

(ii) $1,2$-Dimethylcyclopentane

Here the methyl groups are on adjacent carbons. Each substituted carbon can be stereogenic. There are cis and trans forms.

• cis-1,2-dimethylcyclopentane: both methyls on same side
• trans-1,2-dimethylcyclopentane: methyls on opposite sides

Now check optical activity:

• One of these has a plane of symmetry and is meso $
• The other exists as an enantiomeric pair

For cyclopentane with identical substituents:

• cis-1,2 is meso (achiral) → $1$ stereoisomer
• trans-1,2 is chiral → $2$ stereoisomers

Total for $1,2$: $1+2=3$

(iii) $1,3$-Dimethylcyclopentane

Again, there are cis and trans forms.

For identical substituents in cyclopentane:

• cis-1,3 is chiral → enantiomeric pair → $2$
• trans-1,3 is meso (achiral) → $1$

Total for $1,3$: $2+1=3$

3. Add all stereoisomers

So the total number of possible stereoisomers of dimethylcyclopentane is $\boxed{7}$

4. Compare with stored correct answer

Stored correct answer = $6$

My derived answer = $7$

The likely reason for the discrepancy is that sometimes one positional isomer may be overlooked, but for dimethylcyclopentane the three distinct constitutional isomers $1,1$, $1,2$, and $1,3$ must all be counted. Including their stereoisomers gives $7$ total.