All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is

We need to find the position of the word PUBLIC when all permutations of its letters are arranged in dictionary (alphabetical) order.

The letters of PUBLIC in alphabetical order are: B, C, I, L, P, U.

All 6 letters are distinct, so total arrangements = $$6! = 720$$.

To find the rank of P-U-B-L-I-C:

Position 1: P

Letters before P in the sorted list: B, C, I, L (4 letters).

Words starting with each = $$5! = 120$$.

Count = $$4 \times 120 = 480$$.

Position 2: U (remaining: B, C, I, L, U)

Letters before U: B, C, I, L (4 letters).

Words for each = $$4! = 24$$.

Count = $$4 \times 24 = 96$$.

Position 3: B (remaining: B, C, I, L)

Letters before B: none. Count = $$0$$.

Position 4: L (remaining: C, I, L)

Letters before L: C, I (2 letters).

Count = $$2 \times 2! = 4$$.

Position 5: I (remaining: C, I)

Letters before I: C (1 letter). Count = $$1 \times 1! = 1$$.

Position 6: C

Count = $$0$$.

Rank = $$480 + 96 + 0 + 4 + 1 + 0 + 1 = 582$$.

The correct answer is Option D: 582.