Step 1: Understanding the Concept:
To find the rank of a word in a dictionary, we list words alphabetically and count how many come before the target word 'HELAND'.
The letters in 'HANDLE' in alphabetical order are: A, D, E, H, L, N.
Step 2: Detailed Explanation:
Total letters = 6. Target word: H E L A N D.
1. Words starting with A:
Fix A at the first position. Remaining 5 letters can be arranged in \( 5! \) ways.
\( 1 \times 5! = 120 \).
2. Words starting with D:
Fix D at the first position. Remaining 5 letters can be arranged in \( 5! \) ways.
\( 1 \times 5! = 120 \).
3. Words starting with E:
Fix E at the first position. Remaining 5 letters can be arranged in \( 5! \) ways.
\( 1 \times 5! = 120 \).
4. Words starting with H:
The target word starts with H, so we lock H. Remaining letters to order: A, D, E, L, N.
- Start with HA: (Next available alphabetical letter is A)
Remaining 4 letters: \( 4! = 24 \).
- Start with HD:
Remaining 4 letters: \( 4! = 24 \).
- Start with HE: (Matches target)
Lock E. Remaining letters: A, D, L, N.
- Start with HEA:
Remaining 3 letters: \( 3! = 6 \).
- Start with HED:
Remaining 3 letters: \( 3! = 6 \).
- Start with HEL: (Matches target)
Lock L. Remaining letters: A, D, N.
- Start with HELA: (Matches target)
Lock A. Remaining letters: D, N.
Alphabetical order for remaining: D, then N.
- HELADN: This is the first word. (Rank + 1)
- HELAND: This is the next word. (Rank + 1)
Summing up:
Rank = (Words starting A, D, E) + (HA, HD) + (HEA, HED) + (HELADN) + 1
\[ \text{Rank} = (3 \times 120) + (2 \times 24) + (2 \times 6) + 1 + 1 \]
\[ = 360 + 48 + 12 + 2 \]
\[ = 422 \]
Step 4: Final Answer:
The rank of the word 'HELAND' is 422.