Question

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

Answer 1

Consider the word $EAMCET$. Arranging the letters in alphabetical order results in
$ACEEMT$. This will be the first word in the dictionary. Thus keeping $A$ fixed, the rest of the letters can be re-arranged in $\frac{5!}{2!}$ or $60$ ways.
Now let the first Letter be $C$, the rest of the letters can be re-arranged in $\frac{5!}{2!}$ or $60$ ways.
Now let the first letter be $E$, therefore the first word in the dictionary which starts with $E$ will be $EACEMT$. Let the third letter be $C$, the rest three letters can be re-arranged in $3!$ or $6$ ways. Similarly for $E$ being the third letter, $6$ ways. Now let $M$ be the third letter, thus the first word will be $EAMCET$.
Hence rank of the word EAMCET will be $60+60+6+6+1=120+12+1=133$