Question

All the letters of the word LETTER are arranged in all possible ways and the words (with or without meaning) thus formed are arranged in dictionary order. Then the rank of the word TETLER is:

• A. \(171\)
• B. \(138\)
• C. \(141\)
• D. \(168\)

Hint:

To find the dictionary rank of a word with repeated letters, sort the letters alphabetically and fix one letter at a time, counting permutations of remaining letters considering repetitions.

Answer 1

The Correct Option is C

Step 1: The word is LETTER. Letters in alphabetical order: E, E, L, R, T, T
Step 2: Count total number of permutations:
\[ \text{Total letters} = 6, \quad \text{with E and T repeating twice each} \]
\[ \Rightarrow \text{Total permutations} = \frac{6!}{2! \cdot 2!} = \frac{720}{4} = 180 \]
Step 3: Find the rank of TETLER:
We fix letters one by one and count how many permutations come before "TETLER".
[--] Fix first letter < T: E, L, R → try each:
E: Remaining = E, L, R, T, T → \(\frac{5!}{2!} = 60\)
L: Remaining = E, E, R, T, T → \(\frac{5!}{2! \cdot 2!} = 30\)
R: Remaining = E, E, L, T, T → \(\frac{5!}{2! \cdot 2!} = 30\)
Total before T = \(60 + 30 + 30 = 120\)
[--] Fix T as 1st letter.
Now check 2nd letter < E: only E comes next, so we proceed.
[--] Fix 2nd letter = E. Remaining: E, L, R, T
[--] Fix 3rd letter < T: E, L, R
E: Remaining = L, R, T → \(3! = 6\)
L: Remaining = E, R, T → \(3! = 6\)
R: Remaining = E, L, T → \(3! = 6\)
Total before TET... = \(6 + 6 + 6 = 18\)
[--] Fix 3rd = T. Remaining: E, L, R
[--] Fix 4th < L: E → Remaining = L, R → \(2! = 2\)
[--] Fix 4th = L. Remaining = E, R
[--] Fix 5th < E: none
[--] Fix 5th = E. Remaining = R
[--] 6th = R
Only this matches TETLER
\[ \text{Rank} = 120 (\text{from earlier T cases}) + 18 + 2 + 1 = \boxed{141} \]