Question

If the letters of the word $MASTER$ are permuted in all possible ways and the words thus formed are arranged in dictionary order, then find the rank of the word $MASTER$.

Answer 1

Let us arrange the word $MASTER$ in alphabetic order
$AEMRST$
Thus as per dictionary rank it is the first word.
Now keeping $A$ fixed, the rest of the $5$ words can be re-arranged in $5!$ ways.
$=120$.
Thus the rank of the word starting with $E$ will be greater than $120$.
Now keeping $E$ as the first letter, the rest of the words can be re-arranged in $5!$ ways or $120$ ways.
Now, let us fix $M$ as the first letter. Rearranging the next letters in alphabetic order gives
$MAERST$. Keeping $E$ as the third letter, $RST$ can be permuted in $3!$ or $6$ ways.
The next letter to occupy third position is $R$
$MAREST$. Again $3!=6$ ways. The next letter to occupy the third position is $S$, $MASERT$. Let $E$ be the fourth letter, the remaining letters can be permuted in $2$ ways. Then comes, $MASRET$, again $2$ ways.

Then comes $MASTER$.

Hence the rank is $120+120+6+6+2+2+1=257$.