Question

All the letters of the word MOTHER are arranged in all possible ways and the resulting words (may or may not have meaning) are arranged as in the dictionary. The number of words that appear after the word MOTHER is

• A. 309
• B. 310
• C. 410
• D. 411

Hint:

To find the rank of a word, count how many words come before it in alphabetical order. Count words starting with letters smaller than the first letter, then fix the first letter and count words starting with second letters smaller than the word's second letter, and so on.

Answer 1

The Correct Option is D

Step 1: Total number of arrangements of letters in MOTHER.
The word MOTHER has 6 distinct letters. Hence, total number of words = $6! = 720$.

Step 2: Find the rank of MOTHER in dictionary order.
Alphabetical order: E, H, M, O, R, T.
- Words starting with E: $5! = 120$
- Words starting with H: $5! = 120$
Now consider words starting with M (MOTHER starts with M). Remaining letters: O, T, H, E, R (alphabetical order: E, H, O, R, T).
- Words starting with ME: $4! = 24$
- Words starting with MH: $4! = 24$
- Words starting with MO: remaining letters E, H, R, T
- MOE: $3! = 6$
- MOH: $3! = 6$
- MOR: $3! = 6$
- MOTE: remaining letters E, R → $2! = 2$
- MOTHE → last letter R → MOTHER itself, count = 1
Rank of MOTHER = $120 + 120 + 24 + 24 + 6 + 6 + 6 + 2 + 1 = 309$

Step 3: Words after MOTHER.
Words after = $720 - 309 = 411$