# If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary

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If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E ?

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Given : word EXAMINATION

number of letters = 11

the dictionary words  before starting letter E will be starting with A

Case(1) first letter A and second letter also A

remaining 9 letters have two I and two N

so these are permutations with repetition,

so number of ways = 9!/(2!2!) = 9!/4

Case(2) first letter A and second letter also I

remaining 9 letters have two N

so these are permutations with repetition,

so number of ways = 9!/(2!) = 9!/2

Case(3) first letter A and second letter is N

remaining 9 letters have two I

so these are permutations with repetition,

so number of ways = 9!/(2!) = 9!/2

Case(4) first letter A and second letter is any of the 5 letters E, M, T, O, X

these 5 letters can be chosen in 5 ways

remaining 9 letters have two I and two N

so these are permutations with repetition,

so number of ways = 9!/(2!2!) = 5(9!/4)

The total number of ways from all the 4 cases

= (9!/4) + (9!/2) + (9!/2) + 5(9!/4)

= 9!(1/4 + 1/2 + 1/2 + 5/4)

=9! (5/2)