Question

60 words can be made using all the letters of the word BHBJO, with or without meaning. If these words are written as in a dictionary, then the 50^th word is :

• A. OBBHJ
• B. HBBJO
• C. OBBJH
• D. JBBOH

Answer 1

The Correct Option is C

To find the 50th word in the dictionary order of permutations of the letters in BHBJO, we proceed as follows:

Step 1: Arrange the letters alphabetically

The letters in the word BHBJO, arranged in alphabetical order, are:

\[ B, B, H, J, O. \]

Step 2: Total number of permutations

The total number of permutations of these letters is:

\[ \frac{5!}{2!} = \frac{120}{2} = 60 \quad (\text{since there are 2 repeated } B's). \]

Step 3: Determine the block sizes

Fix the first letter in alphabetical order and calculate the number of permutations for each block.

Case 1: First letter \(B\)

If the first letter is \(B\), the remaining letters are \(B, H, J, O\). The number of permutations of these is:

\[ \frac{4!}{2!} = \frac{24}{2} = 12. \]

Thus, the first 12 words start with \(B\).

Case 2: First letter \(H\)

If the first letter is \(H\), the remaining letters are \(B, B, J, O\). The number of permutations of these is:

\[ \frac{4!}{2!} = \frac{24}{2} = 12. \]

Thus, the next 12 words (from 13 to 24) start with \(H\).

Case 3: First letter \(J\)

If the first letter is \(J\), the remaining letters are \(B, B, H, O\). The number of permutations of these is:

\[ \frac{4!}{2!} = \frac{24}{2} = 12. \]

Thus, the next 12 words (from 25 to 36) start with \(J\).

Case 4: First letter \(O\)

If the first letter is \(O\), the remaining letters are \(B, B, H, J\). The number of permutations of these is:

\[ \frac{4!}{2!} = \frac{24}{2} = 12. \]

Thus, the next 12 words (from 37 to 48) start with \(O\).

Step 4: Focus on the 49th to 60th words

The 49th to 60th words start with \(OB\), since the first letter is \(O\) and the second letter must now be \(B\). The remaining letters to permute are \(B, H, J\). These permutations are:

\[ OBBHJ, OBBJH, OBHBJ, OBJHB, OBJBH, OBJHB. \]

The second word in this list is the 50th word: OBBJH.