Numbers of letters in 'RANDOM' = (R → 1, A → 1, N → 1, D → 1, O → 1, M → 1) = 6
Number of words that begin with A = 
Number of words that begin with D = 
Number of words that begin with M = 
Number of words that begin with O = 
Number of words that begin with N = 
So far, we have formed 120 + 120 + 120 + 120 + 120 = 600 words.
Now words starting with R.
Number of words starting with RAD = 1 x 1 x 1 x 3! = 6.
Number of words starting with RAM = 1 x 1 x 1 x 3! = 6.
Total words = 600 + 12 = 612.
613th word is RANDMO
614th word is RANDOM
∴ Rank of the word random in a dictionary is 614.
Tie the two
Number of arrangements = 
Mix with remaining
Now, we have 3 + 1 = 4 persons to sit around a round table.
The number of permutations = (4 - 1)! = 3! = 6
Hence, the total number of arrangments when the two persons sit together
= 2 x 6 = 12
Total letters in word 'AGAIN' (A → 2, G → 1, I → 1, N → 1) = 5
Number of words that begin with A = 
(Fix A at first place, so that the remaining 4 letters are different)
Number of words with begin with G = 
Number of words that begin with I = 
Thus, so far we have formed 24 + 12 + 12 = 48 words.
The 49th word will begin with N and it is NAAGI and the 50th word is NAAIG.
How many different words can be formed by using the letters of the word ‘ALLAHABAD?
(a) In how many of these do the vowels occupy even positions.
(b) In how many of these, the two L’s do not come together?
Number of letters in word 'ALLAHABAD' = (A → 4, L → 2, H → 1, B → 1, D → 1) = 9
Number of arrangements = 
(a) There are only four A's as vowels.
They can occupy even places (2, 4, 6, 8) in 
ways
∴ Number of ways in which vowels occupying even places = 1
We are left with 5 places and letters (L → 2, H → 1, B → 1, D → 1).
Number of permutations = 
Hence, total number of arrangements in which A's occupy even places
= 1 x 60 = 60.
(b) We first find the number of arrangements in which two L's are not together:
Number of arrangements in which two L's are together
Hence, the number of arrangements in which the two L's are not together
= (Total arrangements) - (the number of arrangements in which the two L's are together)
= 7560 - 1680 = 5880.
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