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".

  1. 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": $$1$$

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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