Question

The letters of the word $SURITI$ are written in all possible orders and these words are written out as in a dictionary. Let k be the rank of the word $SURITI$.Find sum of digits of k ?

Answer 1

The letters of $SURITI$ can be arranged in $\frac{6!}{2!}=369$ ways.
The alphabetic order is $I,R,S,T$ and $U$. The number of words beginning with $I$ is $5!=120$ and those beginning with $R$ is $\frac{5!}{2!}=60$.
Then come words beginning with $SI$ is $4!=24$, $SR$ is $\frac{4!}{2!}=12$ and $ST$ is $\frac{4!}{2!}=12$.
We get ${}^3{P}_3=3!=6$ words beginning with $SUI$, after which comes the words $SURIIT$ and $SURITI$.
Thus the rank of $SURITI$ is $120+60+24+12+12+6+2=236$
Sum of digit is $11$