20. Expand the polynomial $(a+b+c+d+1)^{N}(N \in$ $\mathbf{Z}_{+}$) and combine like terms, resulting in 1001 terms, where each term includes $a, b, c, d$ raised to positive integer powers. Then the value of $N$ is ( ).
(A) $9 \quad$ (B) 14
(C) 16
(D) 17
(E) 19
20. B.

Let each term in the expansion contain $a^{x} b^{y} c^{z} d^{w} 1^{t}$. Then $x+y+z+w+t=N$, where $x, y, z, w \in \mathbf{Z}_{+}, t \in \mathbf{N}$.
Let $x^{\prime}=x-1, y^{\prime}=y-1, z^{\prime}=z-1, w^{\prime}=w-1$.
Thus, $x^{\prime}+y^{\prime}+z^{\prime}+w^{\prime}+t=N-4$.
Notice that, the number of non-negative integer solutions to the indeterminate equation
$$
x^{\prime}+y^{\prime}+z^{\prime}+w^{\prime}+t=N-4
$$

is the same as the number of terms in the expansion of the polynomial $(a+b+c+d+1)^{N}$.
Therefore, $\mathrm{C}_{N}^{4}=1001 \Rightarrow N=14$.