We need the dictionary rank of the word PUBLIC among all permutations of its letters.

The letters are all distinct: $B, C, I, L, P, U$

We count how many valid words come before PUBLIC in dictionary order, then add 1.

1. First letter: P

Letters smaller than P are: $B, C, I, L$ There are $4$ such letters.

For each such choice, the remaining $5$ letters can be arranged in: $5! = 120$ ways.

So words before those starting with P: $4 \times 120 = 480$

2. Second letter: U

Now first letter is fixed as P. Remaining letters are: $B, C, I, L, U$ Among these, letters smaller than U are: $B, C, I, L$ There are $4$ such letters.

For each such choice, remaining $4$ letters can be arranged in: $4! = 24$ ways.

So additional words before PU...: $4 \times 24 = 96$

Running total: $480 + 96 = 576$

3. Third letter: B

Now prefix is PU. Remaining letters are: $B, C, I, L$ Letters smaller than B: none.

So contribution: $0$

Running total: $576$

4. Fourth letter: L

Now prefix is PUB. Remaining letters are: $C, I, L$ Letters smaller than L are: $C, I$ There are $2$ such letters.

For each such choice, remaining $2$ letters can be arranged in: $2! = 2$ ways.

Contribution: $2 \times 2 = 4$

Running total: $576 + 4 = 580$

5. Fifth letter: I

Now prefix is PUBL. Remaining letters are: $C, I$ Letters smaller than I are: $C$ There is $1$ such letter.

For each such choice, remaining $1$ letter can be arranged in: $1! = 1$ way.

Contribution: $1 \times 1 = 1$

Running total: $580 + 1 = 581$

6. Sixth letter: C

No contribution.

So number of words before PUBLIC is: $581$ Hence its serial number is: $581 + 1 = 582$

7. Option check

• A: $578$ ❌
• B: $576$ ❌
• C: $580$ ❌
• D: $582$ ✅

Therefore, the correct answer is: $\boxed{582}$