8. In the expansion of the polynomial $(a+b+c+d)^{8}$, the number of terms where the exponent of each variable is not zero is ( ) terms.
(A) 35
(B) 42
(C) 45
(D) 50
8. A.

Let $(a+b+c+d)^{8}$ have an arbitrary term $p a^{x_{1}} b^{x_{2}} c^{x_{3}} d^{x_{4}}$ in its expansion.
If the exponent of each letter is not zero, then
$$
\begin{array}{l}
x_{i} \geqslant 1(i=1,2,3,4), \text { and } \\
x_{1}+x_{2}+x_{3}+x_{4}=8 . \\
\text { Let } u_{i}=x_{i}-1(i=1,2,3,4) . \text { Then } \\
u_{1}+u_{2}+u_{3}+u_{4}=4 .
\end{array}
$$

Therefore, the number of terms sought is $\mathrm{C}_{7}^{3}=35$.