If the different permutations of all the letter of the words $E X A M I N A T I O N$ 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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Solution

Arranging the words in dictionary order gives us

$A A E I I N N M O T X$

Hence consider the words starting with A

$A - - - - - - - - - -$

The remaining 10 places can be arranged in $\frac{10!}{2! \cdot 2! \cdot 2!}$ ( as there are $2 A, 2 I, 2 N$ )
The next letter that comes is $E .$

Hence the $4, 53, 601^{s t}$ word will be starting with E.

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

Arranging the words in dictionary order gives us AAEIINNMOTXHence consider the words starting with AA−−−−−−−−−−The remaining 10 places can be arranged in 10!2!⋅2!⋅2! ( as there are 2A,2I,2N)The next letter that comes is E.Hence the 4,53,601st word will be starting with E.

If the different permutations of all the letter of the words $E X A M I N A T I O N$ are listed as in a dictionary, how many words are there in this list before the first word starting with $E$ ?