Question

The letter of the word $RANDOM$ are written in all possible orders and these words are written out as dictionary then the rank of the word $RANDOM$ is

Answer 1

Arranging RANDOM in dictionary order gives us
$ADMNOR$.

Consider the words starting with A. If the starting letter is fixed as A, then the remaining 5 letters can be rearranged in 5! ways.

Similarly with the words starting with letters D,M,N,O.
Hence $5\times 5!$
$=5\left(120\right)$
$=600$ ...(i)

Now consider the words starting with R,
$R-----$
In Dictionary order the first word starting with R will be
$RADMNO$
Keeping the positions of RAD fixed we can rearrange the letters MNO in 3! ways.
Similarly for $RAMDNO$.
Hence
$600+2(3!)$
$=612$.

The successive word will be
$RANDMO$ and $RANDOM$.

Hence the rank of the word RANDOM is $612+2$$=614$