Step 1: Read the information.
$X$ is binomial $B(n,p)$ with $p = 0.6$ and mean $E(X) = 6$. We want the variance.
Step 2: Recall the formulas.
For a binomial distribution, mean $= np$ and variance $= npq$, where $q = 1 - p$.
Step 3: Find $q$.
\[ q = 1 - 0.6 = 0.4 \]
Step 4: Notice a shortcut.
We are told $np = E(X) = 6$ already. Variance is $npq = (np)\,q$, so we can use $np = 6$ directly.
Step 5: Compute the variance.
\[ \text{Var}(X) = (np)\,q = 6 \times 0.4 = 2.4 \]
Step 6: Quick check with $n$.
$n = \tfrac{6}{0.6} = 10$, so $npq = 10 \times 0.6 \times 0.4 = 2.4$. Same answer.
\[ \boxed{2.4} \]