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Step 1: Identify the Letters and Their Repetitions
The given word is "EXAMINATION" which has 11 letters: E, X, A, M, I, N, A, T, I, O, N.
Noticing the repetitions:
A appears 2 times
I appears 2 times
N appears 2 times
E, X, M, T, and O appear once each
Step 2: Determine Order of Letters in a Dictionary
Alphabetically, the distinct letters are A, E, I, M, N, O, T, X. The question asks for the number of permutations that appear before any word beginning with "M". Therefore, we need to count all permutations starting with A, E, and I.
Step 3: Calculate Permutations Starting with "A"
When "A" is fixed in the first position, the remaining letters are E, X, A, M, I, N, T, I, O, N. So we have:
1 A left
2 I's
2 N's
E, X, M, T, O each once
The number of ways to arrange these 10 letters is
$ \frac{10!}{2! \times 2!} $
because we divide by the factorials of identical items (2 A's, 2 I's, and 2 N's, but here one A is already used, so only 2 I's and 2 N's remain repeated).
$ 10! = 3628800. $
Hence,
$ \frac{10!}{2! \times 2!} = \frac{3628800}{4} = 907200. $
Step 4: Calculate Permutations Starting with "E"
Fix "E" first; the remaining letters are X, A, M, I, N, A, T, I, O, N. Here:
2 A's
2 I's
2 N's
X, M, T, O each once
The number of ways to arrange these 10 letters is
$ \frac{10!}{2! \times 2! \times 2!} $
because we have three pairs of identical letters (A, I, and N).
$ 10! = 3628800. $
Thus,
$ \frac{10!}{2! \times 2! \times 2!} = \frac{3628800}{8} = 453600. $
Step 5: Calculate Permutations Starting with "I"
Fix "I"; the remaining letters are E, X, A, M, N, A, T, I, O, N. Now:
2 A's total
1 I left (since one I is used)
2 N's
E, X, M, T, O each once
This scenario is similar to having 10 letters with 2 A's, 2 N's, and a single I repeated once. The count is:
$ \frac{10!}{2! \times 2!} = 907200. $
Step 6: Sum All Permutations Before "M"
Adding them together gives:
$ 907200 \,(A) + 453600 \,(E) + 907200 \,(I) = 2268000. $
Final Answer
Therefore, the number of words in the dictionary order before the first word starting with "M" is
2268000.
