2016 AMC 10A Problem 20

For some particular value of N, when (a+b+c+d+1)^N is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables a, b,c, and d, each to some positive power. What is N?

\textbf{(A) }9 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.