Question

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

• A. 119
• B. 149
• C. 176
• D. 179

Hint:

To find the rank of a word, arrange the letters in lexicographical order and count how many words come before the given word.

Answer 1

The Correct Option is D

We need to find the rank of the word COLLEGE when all permutations of its letters are arranged in dictionary order. Step 1: Count Total Arrangements The word "COLLEGE" consists of the letters: \[ C, O, L, L, E, G, E \] Sorted in alphabetical order: \[ C, E, E, G, L, L, O \] Since there are repeated letters (two E's and two L's), the total number of unique words is: \[ \frac{7!}{2!2!} = \frac{5040}{4} = 1260 \] Step 2: Compute the Rank of "COLLEGE" We determine how many words come before "COLLEGE" by considering words starting with letters before 'C'. - No letters come before 'C'. Now, we fix 'C' as the first letter and count words starting with letters before the second letter in "COLLEGE" (which is 'O'). Words Starting with 'C' We now consider words beginning with 'C' and count cases where the second letter is before 'O'. # (i) Second Letter = 'E' Remaining letters: \( C, E, G, L, L, O \) Possible arrangements: \[ \frac{6!}{2!} = \frac{720}{2} = 360 \] # (ii) Second Letter = 'G' Remaining letters: \( C, E, E, L, L, O \) Possible arrangements: \[ \frac{6!}{2!2!} = \frac{720}{4} = 180 \] Step 3: Words Starting with "CO" Now we fix "CO" and count words where the third letter is before 'L'. - Letters available: \( C, O, E, E, G, L, L \) # (i) Third Letter = 'E' Remaining letters: \( C, O, E, G, L, L \) Possible arrangements: \[ \frac{5!}{2!} = \frac{120}{2} = 60 \] # (ii) Third Letter = 'G' Remaining letters: \( C, O, E, E, L, L \) Possible arrangements: \[ \frac{5!}{2!2!} = \frac{120}{4} = 30 \] Step 4: Words Starting with "COL" Now we fix "COL" and count words where the fourth letter is before 'L'. - Available letters: \( C, O, L, E, E, G, L \) # (i) Fourth Letter = 'E' Remaining letters: \( C, O, L, G, L, E \) Possible arrangements: \[ \frac{4!}{2!} = \frac{24}{2} = 12 \] Step 5: Words Starting with "COLL" Now we fix "COLL" and count words where the fifth letter is before 'E'. - Available letters: \( C, O, L, L, E, E, G \) # (i) Fifth Letter = 'E' Remaining letters: \( C, O, L, L, E, G \) Possible arrangements: \[ \frac{3!}{1!} = 6 \] Step 6: Words Starting with "COLLE" Now we fix "COLLE" and count words where the sixth letter is before 'G'. - Available letters: \( C, O, L, L, E, G, E \) # (i) Sixth Letter = 'E' Remaining letters: \( C, O, L, L, E, E, G \) Possible arrangements: \[ 1 \] Final Step: Compute Rank Adding up all previous cases: \[ 36 + 18 + 60 + 30 + 12 + 6 + 1 = 179 \] Final Answer: \(\boxed{179}\) \bigskip