Question: Could you prompt me, please, is it true?

Expected value of $XYZ$, $E(XYZ)$, is not always $E(X)E(Y)E(Z)$, even if $X$, $Y$, $Z$ are not correlated in pairs, because if $X$, $Y$, $Z$ are not correlated in pairs it doesn't entail that they are uncorrelated in aggregate (it is my idea)?

Answer: 
Suppose
$$
(X,Y,Z) = \begin{cases}
(1,1,0) & \text{with probability }1/4 \\
(1,0,1) & \text{with probability }1/4 \\
(0,1,1) & \text{with probability }1/4 \\
(0,0,0) & \text{with probability }1/4
\end{cases}
$$
Then $X,Y,Z$ are pairwise independent, and $E(X)E(Y)E(Z)=1/8\ne 0 = E(XYZ)$.
Let $\Omega=\{\omega\_1,\omega\_2, \omega\_3, \omega\_4\}$, $\mathcal{F}=2^{\Omega}$ and $\mathbb{P}(\{\omega\_i\})=1/4$ for all $i$. It is easy to check that desired random variables are
$$
X=1\_{\{\omega\_1,\omega\_2\}},\quad Y=1\_{\{\omega\_1,\omega\_3\}},\quad Z=1\_{\{\omega\_1,\omega\_4\}}
$$