JEE Main 2023 — Mathematics Algebra

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$=5!\times 4=480$=5!×4=480

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

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

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

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

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

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

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

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

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

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