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?
907200
Explanation
This question follows the same logic as Example 22 in the textbook. When words are listed in dictionary order, they are arranged alphabetically. Since A comes before E in the alphabet, all words starting with A will appear before any word starting with E. We need to count all permutations starting with A.
Solution Steps
Step 1: Identify the letters in EXAMINATION. The word has 11 letters: E, X, A, M, I, N, A, T, I, O, N. Here, A appears 2 times, I appears 2 times, N appears 2 times, and E, X, M, T, O each appear once.
Step 2: Determine which letters come before E in dictionary order. In alphabetical order, only A comes before E. So all words starting with A will appear before the first word starting with E.
Step 3: Calculate the number of words starting with A. Fix A at the first position. The remaining 10 letters are: E, X, A, M, I, N, T, I, O, N.
Step 4: Count arrangements of remaining letters. In the remaining 10 letters: A appears 1 time, I appears 2 times, N appears 2 times, and E, X, M, T, O each appear once.
Step 5: Apply the permutation formula. Number of words starting with A = 10! / (2! × 2!) = 3628800 / 4 = 907200.