QJEEM24Ma2996: JEE Main 2024 Mathematics - Class 11 JEEM24Ma2996 Solution: 60 words can be made using all the letters of the word , with…

To find the $50^{\text{th}}$ word formed by the letters of "BHBJO" as if listed in a dictionary, let's analyze the arrangement methodically. Given the letters are B, H, B, J, O, there are some repetitions with the letter B appearing twice.

First, calculate the total number of permutations of these letters:

$\frac{5!}{2!} = 60$

Let's arrange the letters in alphabetical order first: B, B, H, J, O.

We need to systematically count the words while following dictionary order:

1. Words starting with B:

• Next position letters: B, H, J, O
• Number of permutations: $\frac{4!}{1!} = 24$ words

Since 24 words starting with 'B' exist and are less than 50, Move to next starting letter alphabetically.

2. Words starting with H:

• Next position letters: B, B, J, O
• Number of permutations: $\frac{4!}{2!} = 12$ words

After 'B', tally becomes 24 (B-words) + 12 (H-words) = 36, needs more.

3. Words starting with J:

• Next position letters: B, B, H, O
• Number of permutations: $\frac{4!}{2!} = 12$ words

Now tally is 36 + 12 = 48 words.

Still 2 more to reach 50.

4. Words starting with O:

• Next position letters: B, B, H, J
• Number of permutations: $\frac{4!}{2!} = 12$ words
• Our answer must be here since 48 + 2 more = 50 total.

First permutation: $OBB H J$

Second permutation (50th word): $OBB J H$

So, the $50^{\text{th}}$ word is: OBBJH

Thus, the correct answer is Option A: OBBJH.