Find the rank of the word LATE, if the letters of the word LATE are permuted and words so formed are arranged

Exercise 6.3 · Q12

Q. Find the rank of the word LATE, if the letters of the word LATE are permuted and words so formed are arranged as in dictionary.

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List words in dictionary (alphabetical) order by fixing letters position by position, counting how many precede LATE.

Dictionary-rank method: for each position (left to right), count how many available letters are alphabetically smaller than the actual letter used there; each such smaller letter contributes (number of arrangements of the remaining letters) words that come earlier.

Letters of LATE: L, A, T, E — sorted alphabetically: A, E, L, T.
1st position of LATE is L. Letters alphabetically before L (among A,E,L,T): A, E — that's 2 letters. Each, if placed first, allows the remaining 3 letters to be arranged in $3!=6$ ways. Words before those starting with L: $2\times3!=2\times6=12$ .
2nd position: with 1st letter fixed as L, remaining letters to place are A, E, T (sorted: A, E, T). Actual 2nd letter of LATE is A, which is the smallest of these — 0 letters come before it. …

Q.Find the rank of the word LATE, if the letters of the word LATE are permuted and words so formed are arranged as in dictionary.