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Step-by-Step Solution
Step 1: Arrange the letters of “MONDAY” in alphabetical order
Write the letters in ascending alphabetical order: A, D, M, N, O, Y.
We will consider all possible six-letter arrangements of these letters as if they were listed in dictionary (lexicographic) order.
Step 2: Count words beginning with letters that come before M
(a) Words starting with A: With A as the first letter, the remaining 5 letters (D, M, N, O, Y) can be arranged in
$5! = 120$ ways.
(b) Words starting with D: With D as the first letter, the remaining 5 letters (A, M, N, O, Y) can also be arranged in
$5! = 120$ ways.
Thus, total words starting with A or D = $120 + 120 = 240$.
Step 3: Count words beginning with M and second letter less than O
When the first letter is M, the remaining letters are A, D, N, O, Y. We now look for cases where the second letter is A, D, or N (each of which is less than O alphabetically).
(a) MA: Fix MA as the first two letters. The remaining 4 letters (D, N, O, Y) can be arranged in
$4! = 24$ ways.
(b) MD: Fix MD as the first two letters. The remaining 4 letters (A, N, O, Y) can be arranged in
$4! = 24$ ways.
(c) MN: Fix MN as the first two letters. The remaining 4 letters (A, D, O, Y) can be arranged in
$4! = 24$ ways.
Total for these three scenarios = $24 + 24 + 24 = 72$.
Adding these to the previous total: $240 + 72 = 312$.
Step 4: Count words beginning with MO but third letter less than N
Next, we look at words that start with MO. The remaining letters (A, D, N, Y) are considered, and we find third letters that come before N in alphabetical order: A and D.
(a) MOA: Fix MOA as the first three letters. The remaining 3 letters (D, N, Y) can be arranged in
$3! = 6$ ways.
(b) MOD: Fix MOD as the first three letters. The remaining 3 letters (A, N, Y) can also be arranged in
$3! = 6$ ways.
Total in these two cases = $6 + 6 = 12$.
Adding to our current total: $312 + 12 = 324$.
Step 5: Words beginning with MON and finding “MONDAY”
Now we look at words starting with MON. The remaining letters after removing M, O, N are A, D, Y.
(a) MONA: Fixing MONA as the first four letters, the remaining two letters (D, Y) can be arranged in
$2! = 2$ ways: “MONADY” and “MONAYD.” Both precede “MONDAY” in alphabetical order.
So we add these 2 words to our count: $324 + 2 = 326$.
The next arrangement in the dictionary order is “MONDAY.” Hence, “MONDAY” is the
$327^\text{th}$ word in the list.
Answer: 327
