4. Given the sequences $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$, their partial sums up to the $n$-th term are $A_{n}$ and $B_{n}$ respectively. Let $c_{n}=a_{n} B_{n}+b_{n} A_{n}-a_{n} b_{n}(n \geqslant 1)$. Then the sum of the first 10 terms of the sequence $\left\{c_{n}\right\}$ is ( ).
(A) $A_{10}+B_{10}$
(B) $\frac{A_{10}+B_{10}}{2}$
(C) $A_{10} B_{10}$
(D) $\sqrt{A_{10} B_{10}}$
4.C.

When $n \geqslant 2$,
$$
\begin{array}{l}
c_{n}=\left(A_{n}-A_{n-1}\right) B_{n}+\left(B_{n}-B_{n-1}\right) A_{n}- \\
\quad\left(A_{n}-A_{n-1}\right)\left(B_{n}-B_{n-1}\right) \\
=A_{n} B_{n}-A_{n-1} B_{n-1} . \\
\text { Therefore, } c_{1}+c_{2}+\cdots+c_{10} \\
=A_{1} B_{1}+\left(A_{2} B_{2}-A_{1} B_{1}\right)+\left(A_{3} B_{3}-A_{2} B_{2}\right)+\cdots \\
+\left(A_{10} B_{10}-A_{9} B_{9}\right)=A_{10} B_{10} .
\end{array}
$$