1. Identify what kind of compounds are possible

A substituted benzene derivative has a benzene ring, i.e. a phenyl part contributing $\mathrm{C_6H_5}$ when one hydrogen is replaced.

Given molecular formula: $\mathrm{C_9H_{12}}$

Compare with benzene: $\mathrm{C_6H_6}$

So the substituents together must contribute:

• Carbon increase: $9-6=3$
• Hydrogen increase: $12-6=6$

Thus we need benzene derivatives containing 3 extra carbons total.

Also, degree of unsaturation check: $\text{IHD} = \frac{2(9)+2-12}{2}=\frac{8}{2}=4$ A benzene ring itself accounts for 4, so the side chain(s) must be saturated and acyclic.

So we only need to distribute 3 carbons as alkyl substituents on a benzene ring.

2. Possible ways to distribute 3 carbons among substituents

The partitions of 3 are:

1. $3$
2. $2+1$
3. $1+1+1$

We count structural isomers for each case.

3. Case 1: One 3-carbon substituent

This gives propylbenzene type compounds.

A 3-carbon alkyl group can attach to benzene in two different ways:

• n-propylbenzene: $\mathrm{C_6H_5CH_2CH_2CH_3}$
• isopropylbenzene (cumene): $\mathrm{C_6H_5CH(CH_3)_2}$

So, number of isomers in this case: $2$

4. Case 2: One ethyl and one methyl substituent

This is ethylmethylbenzene.

For a disubstituted benzene with two different substituents, the distinct positional isomers are:

• ortho $(1,2)$
• meta $(1,3)$
• para $(1,4)$

So we get:

• $o$-ethyltoluene
• $m$-ethyltoluene
• $p$-ethyltoluene

Hence number of isomers in this case: $3$

5. Case 3: Three methyl substituents

This gives trimethylbenzene.

Distinct positional arrangements of three identical methyl groups on benzene are:

• $1,2,3$-trimethylbenzene
• $1,2,4$-trimethylbenzene
• $1,3,5$-trimethylbenzene

So number of isomers in this case: $3$

6. Total number of structural isomers

Adding all cases: $2+3+3=8$

So the total number of structural isomers is: $\boxed{8}$

7. Comparison with stored answer

Stored correct answer = $8$

Our derived answer = $8$

Hence, they agree.