Let $X_1, X_2, X_3 \sim$ i.i.d. $X$, where $E[X]=\mu, \operatorname{Var}(X)=\sigma^2$
$$
\begin{aligned}
\operatorname{Var}(A) & =\operatorname{Var}\left(\frac{X_1+X_2+X_3}{3}\right) \\
& =\frac{1}{9}\left(\operatorname{Var}\left[X_1\right]+\operatorname{Var}\left[X_2\right]+\operatorname{Var}\left[X_3\right]\right) \\
& =\frac{1}{9}\left(3 \sigma^2\right)=\frac{\sigma^2}{3}
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{Var}(B) & =\operatorname{Var}\left(0.1 X_1+0.3 X_2+0.6 X_3\right) \\
& =0.01 \operatorname{Var}\left[X_1\right]+0.09 \operatorname{Var}\left[X_2\right]+0.36 \operatorname{Var}\left[X_3\right] \\
& =0.46 \sigma^2
\end{aligned}
$$
$$
\begin{aligned}
\operatorname{Var}(C) & =\operatorname{Var}\left(0.2 X_1+0.3 X_2+0.5 X_3\right) \\
& =0.04 \operatorname{Var}\left[X_1\right]+0.09 \operatorname{Var}\left[X_2\right]+0.25 \operatorname{Var}\left[X_3\right] \\
& =0.38 \sigma^2
\end{aligned}
$$
Therefore, $\operatorname{Var}(B) \geq \operatorname{Var}(C) \geq \operatorname{Var}(A)$.
