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: Sort the letters of MOTHER alphabetically.
The letters E, H, M, O, R, T are already in the alphabetical order we need.

Step 2: Count words that start with an earlier letter, one position at a time.
Position 1 (M): letters E, H come before M, each contributing \(5!=120\) arrangements of the rest, giving \(2\times120=240\).
Position 2, after fixing M (O): among the remaining letters E, H, O, R, T, the ones before O are E, H, each contributing \(4!=24\), giving \(2\times24=48\).
Position 3, after fixing MO (T): among the remaining E, H, R, T, the ones before T are E, H, R, each contributing \(3!=6\), giving \(3\times6=18\).
Position 4, after fixing MOT (H): among the remaining E, H, R, the only one before H is E, contributing \(2!=2\), giving \(1\times2=2\).
Position 5, after fixing MOTH (E): the remaining letters are E and R, and E is already the smaller one, so no extra words are added here.

Step 3: Add everything and count the word itself.
\[ 240+48+18+2+0+1 = 309 \]

Step 4: Conclusion.
\[ \boxed{309} \]