If the different permutations of all the letter of the word $\mathrm{EXAMINATION}$ are listed as in a dictionary, how many words are there in this list before the first word starting with $\mathrm{E}$ ?
If the different permutations of all the letter of the word $\mathrm{EXAMINATION}$ are listed as in a dictionary, how many words are there in this list before the first word starting with $\mathrm{E}$ ?
In the given word $EXAMINATION$, there are $11$ letters out of which, $A ,$ $I$ and $N$ appear $2$ times and all the other letters appear only once.
The words that will be listed before the words starting with $E$ in a dictionary will be the words that start with $A $only.
Therefore, to get the number of words starting with $A$, the letter $A$ is fixed at the extreme left position, and then the remaining $10$ letters taken all at a time are rearranged.
since there are $2$ Is and $2$ $Ns$ in the remaining $10$ letters,
Number of words starting with $A=\frac{10 !}{2 ! 2 !}=907200$
Thus, the required numbers of words is $907200 .$
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