Question

What is the rank of the word ``MOTHER'' when all possible words are formed using all its letters and arranged in dictionary order?

• A. 308
• B. 309
• C. 291
• D. 307

Hint:

Use dictionary order by counting permutations of letters smaller than the current letter at each position to find the rank of a word.

Answer 1

The Correct Option is B

Step 1: List letters alphabetically.
Letters of ``MOTHER'' in order: E, H, M, O, R, T

Step 2: Count words starting with letters before M.
- Letters before M: E, H
- Each choice has \(5! = 120\) arrangements of remaining letters.
- Total words before M: \(2 \cdot 120 = 240\)

Step 3: Consider words starting with M.
Next letter in MOTHER is O. Letters before O among remaining letters: E, H
- For each of these letters, remaining 4 letters: \(4! = 24\)
- Total words before MO: \(2 \cdot 24 = 48\)

Step 4: Consider words starting with MO.
Next letter in MOTHER is T. Letters before T among remaining letters: E, H, R
- For each of these letters, remaining 3 letters: \(3! = 6\)
- Total words before MOT: \(3 \cdot 6 = 18\)

Step 5: Consider words starting with MOT.
Next letter in MOTHER is H. Letters before H among remaining letters: E
- Remaining 2 letters can be arranged in \(2! = 2\) ways
- Total words before MOTH: \(1 \cdot 2 = 2\)

Step 6: Final rank.
Add 1 for the word itself: \[ \text{Rank} = 240 + 48 + 18 + 2 + 1 = 309 \]