Question

By using all the letters of the word 'LETTER', all possible 6-letter words (with or without meaning) are formed. If all these words are arranged in dictionary order, then the rank of the word 'TRELET' is:

• A. \( 159 \)
• B. \( 147 \)
• C. \( 135 \)
• D. \( 157 \)

Hint:

Always list the unique letters in alphabetical order first. Be extremely careful with repetitions (E and T appear twice); forgetting to divide by \( 2! \) for repeated letters is the most common error in rank problems.

Answer 1

The Correct Option is A

Concept: To find the rank of a word, we count the number of words that come before it alphabetically.
• Letters of 'LETTER' in alphabetical order: E, E, L, R, T, T.
• Use the formula for permutations with repetition: \( \frac{n!}{p!q!...} \).

Step 1: Counting words starting with E, L, and R.
- Words starting with E: Fixed E, remaining {E, L, R, T, T} can be arranged in \( \frac{5!}{2!} = 60 \) ways. - Words starting with L: Fixed L, remaining {E, E, R, T, T} can be arranged in \( \frac{5!}{2!2!} = 30 \) ways. - Words starting with R: Fixed R, remaining {E, E, L, T, T} can be arranged in \( \frac{5!}{2!2!} = 30 \) ways. Subtotal: \( 60 + 30 + 30 = 120 \).

Step 2: Counting words starting with T.
- Words starting with TE: Fixed TE, remaining {E, L, R, T} can be arranged in \( 4! = 24 \) ways. - Words starting with TL: Fixed TL, remaining {E, E, R, T} can be arranged in \( \frac{4!}{2!} = 12 \) ways. Subtotal: \( 120 + 24 + 12 = 156 \).

Step 3: Finding the final rank.
Now we look at words starting with TR. The alphabetical order is: 1. TREE..: {L, T} gives TREELT (157th) and TREETL (158th). 2. TREL..: The first word is TRELET (159th). Thus, the rank of 'TRELET' is 159.