The letters of the word “OLUVME” are permutated and all possible words so formed and  arranged as in a dictionary with serial numbers, then the serial number of the word  “VOLUME” is

Alphabetical order is E, L, M, O, U, V
In dictionary order,
No. of words begin with E…………………….  is 5!
     L…………………….  is 5!
     M…………………….  is 5!
     O…………………….  is 5!
     U…………………….  is 5!
     V E………………….  is 4!
     V L………………….  is 4!
     V M………………….  is 4!
     V O E………………….  is 3!
     V O L E……………….  is 2!
     V O L M……………….  is 2!
     V O L U E……………….  is 1!
 The next work is  V O L U M E
Total  $=5(5!)+3(4!)+3!+2(2!)+1+1$

$=600+72+6+4+2$

$=684$