Proof Writing & Problem Solving · Superchallenge problem

A proof about four real numbers $a, b, c, d$ begins "Without loss of generality $a \le b \le c \le d$." A reader objects, and the writer instead splits into cases by the complete order pattern of $a, b, c, d$, ties included (for example $a = b < d < c$ is one case). How many cases does that write-up have?