Question

If all letters of the word "CHCJL" be arranged in an English dictionary, what will be the 50th word?

• A. HCCLJ
• B. LCCHJ
• C. LCCJH
• D. JHCLC
• E. None of the above

Hint:

For dictionary order ranking problems, always place letters alphabetically, then count blocks of permutations by fixing the first letter step by step. This avoids manual listing of each arrangement.

Answer 1

The Correct Option is C

Step 1: Arrange letters alphabetically.
The letters are C, H, C, J, L. In alphabetical order: C, C, H, J, L.
So, the very first word in dictionary order would be CCHJL.
Step 2: Count words starting with C.
If first letter = C, then remaining letters = C, H, J, L.
Number of arrangements = 4! = 24 words.
Thus, words 1 to 24 in the dictionary will start with C.
Step 3: Count words starting with H.
If first letter = H, then remaining letters = C, C, J, L.
Number of arrangements = \(\dfrac{4!}{2!} = 12\) words (since two Cs are identical).
Thus, words 25 to 36 in the dictionary will start with H.
Step 4: Count words starting with J.
If first letter = J, then remaining letters = C, C, H, L.
Number of arrangements = \(\dfrac{4!}{2!} = 12\).
Thus, words 37 to 48 will start with J.
Step 5: Find the 49th and 50th words.
After 48 words, the next block starts with L.
If first letter = L, then remaining letters = C, C, H, J.
- The 49th word will be the first arrangement under L, which is LCCHJ.
- The 50th word will be the second arrangement, which is LCCJH.
Step 6: Conclude.
Therefore, the 50th word is LCCJH.
\[ \boxed{LCCJH \; \text{(Option C)}} \]