All 120 arrangements of the letters of the word MEDIA are written out in dictionary order, as in a word list. At what position in that list does the word MEDIA itself stand?

In dictionary order the five letters rank A, D, E, I, M. Count the words that come BEFORE MEDIA, letter by letter. First letter smaller than M — A, D, E or I: $4\times 4$! = 96. With M fixed, second letter smaller than E out of {A, D, E, I} — A or D: $2\times 3$! = 12, running total 108. With ME fixed, third letter smaller than D out of {A, D, I} — only A: $1\times 2$! = 2, total 110. With MED fixed, fourth letter smaller than I out of {A, I} — only A: $1\times 1$! = 1, total 111. The last letter is then forced. So 111 words precede MEDIA, and its position is 111 + 1 = 112. Trap 111 forgets that last step from 'how many come before' to 'which position it occupies'. Trap 96 stops after the first letter and trap 108 after the second.