What is the order of the word 'ZENITH' in the dictionary?
Question
What is the order of the word 'ZENITH' in the dictionary?
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Answer
To determine the order of the word "ZENITH" in the dictionary, follow these steps:
Step 1: Arrange the Letters Alphabetically
First, we need to arrange the letters of "ZENITH" in alphabetical order: $$ \mathrm{E}, \mathrm{H}, \mathrm{I}, \mathrm{N}, \mathrm{T}, \mathrm{Z} $$
Step 2: Calculate Potential Positions of Other Words
Next, we calculate the number of permutations for words starting with each letter other than "Z":
-
Words starting with $\mathrm{E}$, $\mathrm{H}$, $\mathrm{I}$, $\mathrm{N}$, $\mathrm{T}$:
- For each of these letters, there would be $5!$ permutations of the remaining 5 letters.
$$ 5 \times 5! = 5 \times 120 = 600 $$
Step 3: Consider Words Starting with "Z"
Now, let's focus on words starting with "Z".
-
Words starting with $\mathrm{ZE}$:
- Possible words:
- "ZEH***" (where *** are any permutations of I, N, T): $$3! = 3 \times 2 \times 1 = 6$$
- "ZEI***" (where *** are any permutations of H, N, T): $$3! = 3 \times 2 \times 1 = 6$$
- For words starting with "ZEN":
- "ZENH***" (where *** are any permutations of I, T): $$2! = 2 \times 1 = 2$$
- "ZENITH": $$1$$
- Possible words:
Step 4: Summing Up
Summarize all permutations:
- Total words starting with $\mathrm{E}$, $\mathrm{H}$, $\mathrm{I}$, $\mathrm{N}$, $\mathrm{T}$: $$600$$
- Words starting with "Z" and followed by "EH": $$6$$
- Words starting with "Z" and followed by "EI": $$6$$
- Words starting with "ZENH": $$2$$
- The word "ZENITH": $$1$$
Therefore, the total order of the word "ZENITH" is: $$ \begin{aligned} &600 + 6 + 6 + 2 + 1 \ =& 615 \end{aligned} $$
Thus, the word "ZENITH" is the 615th word in the dictionary.
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