The letters of the word RANDOM are written in all

 possible orders and these words are written out as in 

a dictionary, then the rank of the word RANDOM, is  

ln a dictionary the words at each stage are

 arranged in alphabetical order. In the given 

problem we must therefore consider the words

 beginning with $A, D, M, N, 0, R$  in order. A will

 occur in the first place as often as there Me

 \vays of arranging the remainingS letters all at

 a time. i.e. A will occur 5! times. $D, M, N, 0$ will

 occur in the first place the same number   

of times. So, 

Number of words starting with $A =5 ! = 120$

 Number of words starling with  $D =5 ! = 120$

 Number of words starting with $M =5! = 120$

 Number of words starting with N = 5! == 120 .

Number of words starting with $O =5 !=120$ 

Number of words beginning with  $RAD$ $or RAM, is 3!$ 

Now the words beginning with' RAN' must follow  

First one is RANDMO .md the next one is RANDOM . 

  $\therefore$  Rank of RANDOM-"' (5 !) 5+(31) 2 + 2 = 614