All the letters of the word PUBLIC are written in all possible orders and these words are written as in a…

The given word is PUBLIC

Arranging the letters alphabetically, we get

BCILPU

When the word starts with any of the letters B/C/I/L, the number of possibilities$=5!\times 4=480$

Now when the word starts with PB, then the number of possibilities$=4!=24$

Now when the word starts with PC, then the number of possibilities $=4!=24$

Now when the word starts with PI, then the number of possibilities $=4!=24$

Now when the word starts with PL, then the number of possibilities $=4!=24$

Now when the word starts with PUBC, then the number of possibilities$=2!=2$

Now when the word starts with PUBI, then the number of possibilities $=2!=2$

Now when the word starts with PUBLC, then the number of possibilities$=1$

Now when the word starts with PUBLIC, then the number of possibilities $=1$

Rank $=480+24\times 4+2\times 2+1\times 2=582$

Hence, rank of the work PUBLIC is $582$.