The letter of the word SURITI are written in all possible orders and these words are written out as in a dictionary. Then the rank of the word SURITI is

The letters of SURITI can be arranged in $6!/2!=720/2=360$  ways. The alphabetic order is $I,R,S,T$  and $U.$  

The number of words beginning $I$  is ${}^5{P}_5$  $=5!=120,$  and those beginning R number $5!/2!$  $=120/2=60.$  

Then come words beginning

$4!/2!=12.$ We get ${}^3{P}_3=3!=6$  words beginning SUI, after which come the words SURIIT and SURITI.

Thus the rank of SURITI is$120+60+24+12+6+2=224.$