1. Identify mesityl oxide structure

Mesityl oxide is the aldol condensation product of acetone. Its structure is:

$\text{CH}_3-CO-CH=C(\text{CH}_3)_2$

More clearly, it can also be written as:

$\text{CH}_3-CO-CH=C(\text{CH}_3)-\text{CH}_3$

Its molecular formula is indeed $C_6H_{10}O$.

2. Count all C–C sigma bonds

A sigma bond exists in every single bond, and in a double bond there is one sigma + one pi bond.

Now count carbon-carbon connections:

• Between left $\text{CH}_3$ and carbonyl carbon: 1 C–C sigma bond
• Between carbonyl carbon and adjacent $\text{CH}$: 1 C–C sigma bond
• Between $\text{CH}$ and the alkene carbon $C$: this is a C=C double bond, so contributes 1 sigma bond
• Between alkene carbon and one $\text{CH}_3$: 1 C–C sigma bond
• Between alkene carbon and the other $\text{CH}_3$: 1 C–C sigma bond

Total C–C sigma bonds:

$1+1+1+1+1=5$

3. Final answer

$\boxed{5}$

4. Comparison with stored correct answer

Stored correct answer = $5$

My derived answer = $5$

So, they agree.