Question:

Step 1: Identify the letters and calculate the total number of permutations. The given word is BHBJO. The letters are B, B, H, J, O. The total number of distinct words that can be formed using all these letters is given by the permutation formula for multiset, where there are 5 letters in total and the letter 'B' repeats 2 times: $$ \text{Total words} = \frac{5!}{2!} = \frac{120}{2} = 60 $$ This matches the total number of words mentioned in the problem. We need to find the 50th word when arranged in dictionary order. The letters in alphabetical order are B, H, J, O. Step 2: Calculate the number of words starting with B, H, and J. We arrange the letters in alphabetical order: B, H, J, O. * **Words starting with B:** If the first letter is B, the remaining 4 letters are B, H, J, O (one B is used). These 4 distinct letters can be arranged in $4!$ ways. $$ \text{Number of words starting with B} = 4! = 24 $$ These are the 1st to 24th words. * **Words starting with H:** If the first letter is H, the remaining 4 letters are B, B, J, O. Since 'B' repeats 2 times, these letters can be arranged in $\frac{4!}{2!}$ ways. $$ \text{Number of words starting with H} = \frac{4!}{2!} = \frac{24}{2} = 12 $$ These are the 25th to $24+12=36$th words. * **Words starting with J:** If the first letter is J, the remaining 4 letters are B, B, H, O. Since 'B' repeats 2 times, these letters can be arranged in $\frac{4!}{2!}$ ways. $$ \text{Number of words starting with J} = \frac{4!}{2!} = \frac{24}{2} = 12 $$ These are the 37th to $36+12=48$th words. Step 3: Determine the starting letter for the 49th word. From the calculations above, the first 48 words are formed by words starting with B, H, and J. $$ 24 (\text{starting with B}) + 12 (\text{starting with H}) + 12 (\text{starting with J}) = 48 $$ Therefore, the 49th word will be the first word starting with the next letter in alphabetical order, which is O. Step 4: Find the 49th and 50th words among those starting with O. The letters available to form words starting with O are B, B, H, J. To find words in dictionary order, we arrange these remaining letters alphabetically. The alphabetical order of these letters is B, B, H, J. * **For the 49th word:** We take O as the first letter and arrange the remaining letters (B, B, H, J) in their lexicographical order. The first arrangement of B, B, H, J is BBHJ. So, the 49th word is OBBHJ. * **For the 50th word:** We need the next word in dictionary order after OBBHJ. This involves finding the next lexicographical permutation of B, B, H, J. The next permutation of BBHJ is BBJH. So, the 50th word is OBBJH. Step 5: State the final answer. The 50th word in dictionary order is OBBJH. Comparing this with the given options: Option 1: JBBOH Option 2: OBBJH Option 3: OBBHJ Option 4: HBBJO The correct option is OBBJH. The final answer is $\boxed{\text{OBBJH}}$.