2016 AMC 10A Problem 20

A term that includes all four variables has form $a^vb^wc^xd^y$, with $v,w,x,y\ge1$, multiplied by some power of $1$. If that power is $z$, then $$v+w+x+y+z=N.$$ After subtracting $1$ from each of $v,w,x,y$, this becomes a nonnegative stars-and-bars count with $5$ variables summing to $N-4$. The number of such terms is $$\binom{N}{4}.$$ Since $\binom{14}{4}=1001$, we get $N=14$. Thus, the correct answer is B.