JEE Mathematics 2023 Question with Solution

Given: All distinct letters of the word PUBLIC are arranged in dictionary order.

Find: The serial number of the word PUBLIC.

Arrange the letters alphabetically: $B, C, I, L, P, U$.

Count the words that come before PUBLIC.

Words beginning with letters smaller than $P$ are:

So, total before words starting with $P$ is

Now fix the first letter as $P$. Remaining letters are $B, C, I, L, U$.

For the second letter of PUBLIC, we need $U$. Smaller available letters are $B, C, I, L$.

Hence,

So, total before words starting with $PU$ is

Now fix $PU$. Remaining letters are $B, C, I, L$.

For the third letter of PUBLIC, we need $B$. Since $B$ is the smallest among the remaining letters, no word comes before PUB... at this step.

Now fix $PUB$. Remaining letters are $C, I, L$.

For the fourth letter of PUBLIC, we need $L$. Smaller available letters are $C$ and $I$.

Therefore,

So, total before words starting with $PUBL$ is

Now fix $PUBL$. Remaining letters are $I, C$.

For the fifth letter of PUBLIC, we need $I$. Smaller available letter is $C$.

Hence,

is preceded by

word.

Therefore, the rank of PUBLIC is

The correct option is D.