Key Concepts and Formulas

• Permutations with Repetition: The number of permutations of $n$ objects where $n_1$ are alike, $n_2$ are alike, ..., $n_k$ are alike is given by $\frac{n!}{n_1! n_2! ... n_k!}$.
• Dictionary Order: Words are arranged alphabetically, considering the position of each letter.

Step-by-Step Solution

Step 1: Calculate the total number of permutations.

We are given the word BHBJO. There are 5 letters, with the letter B repeated twice. The total number of permutations is: $\frac{5!}{2!} = \frac{120}{2} = 60$ This confirms that there are 60 distinct words that can be formed.

Step 2: Determine the number of words starting with 'B'.

If the first letter is 'B', the remaining letters are B, H, J, O. The number of permutations of these 4 letters is: $\frac{4!}{1!} = 4! = 24$ This means there are 24 words starting with 'B'.

Step 3: Determine the number of words starting with 'H'.

If the first letter is 'H', the remaining letters are B, B, J, O. The number of permutations of these 4 letters is: $\frac{4!}{2!} = \frac{24}{2} = 12$ This means there are 12 words starting with 'H'.

Step 4: Determine the number of words starting with 'J'.

If the first letter is 'J', the remaining letters are B, B, H, O. The number of permutations of these 4 letters is: $\frac{4!}{2!} = \frac{24}{2} = 12$ This means there are 12 words starting with 'J'.

Step 5: Determine the number of words starting with 'O'.

If the first letter is 'O', the remaining letters are B, B, H, J. The number of permutations of these 4 letters is: $\frac{4!}{2!} = \frac{24}{2} = 12$ This means there are 12 words starting with 'O'.

Step 6: Find the range containing the 50th word.

• Words starting with 'B': 24 words (1-24)
• Words starting with 'H': 12 words (25-36)
• Words starting with 'J': 12 words (37-48)
• Words starting with 'O': 12 words (49-60)

Since 48 words start with B, H, or J, the 49th word and the 50th word must start with 'O'.

Step 7: Find the 49th word.

The 49th word is the first word starting with 'O'. The remaining letters in alphabetical order are B, B, H, J. Therefore, the 49th word is OBBHJ.

Step 8: Find the 50th word.

To find the 50th word, we need to find the next permutation after OBBHJ. The letters are O, B, B, H, J. Keeping OBB fixed, we can swap H and J to get OBBJH.

Common Mistakes & Tips

• Remember to divide by the factorial of the number of repetitions when calculating permutations.
• Systematically list the words in dictionary order to avoid errors.
• When looking for the nth word, identify the range of words starting with each letter.

Summary

We found the 50th word by first calculating the number of permutations of the letters in BHBJO. Then, we systematically counted the number of words starting with each letter in alphabetical order until we reached the range containing the 50th word (starting with O). We then found the first word starting with O, OBBHJ, and the next word in dictionary order, OBBJH, which is the 50th word.

Final Answer

The final answer is \boxed{OBBJH}, which corresponds to option (A).