The question presents a sequence of alphanumeric pairs: $B_{3}$, $E_{4}$, $H_{7}$, $K_{8}$, $N_{9}$, ?, $T_{11}$, $W_{13}$, $Z_{24}$. We need to find the pair that replaces the question mark (?).

Pattern Analysis

The sequence can be analyzed by considering the letters and numbers separately.

Letter Progression

• Identify the letters: B, E, H, K, N, ?, T, W, Z.
• Find their positions in the alphabet: 2, 5, 8, 11, 14, ?, 20, 23, 26.
• Calculate the difference between consecutive letter positions:
  • $5 - 2 = 3$
  • $8 - 5 = 3$
  • $11 - 8 = 3$
  • $14 - 11 = 3$
• The pattern for the letters is an addition of 3 to the alphabet position.
• Following this pattern, the missing letter's position is $14 + 3 = 17$.
• The 17th letter of the alphabet is Q.
• Verify the pattern after the missing term: $20 - 17 = 3$ (T - Q), $23 - 20 = 3$ (W - T), $26 - 23 = 3$ (Z - W). The pattern holds.
• Therefore, the missing letter is Q.

Number Progression

• Identify the numbers in the sequence: 3, 4, 7, 8, 9, ?, 11, 13, 24.
• Based on the letter analysis, the missing pair must start with Q. The relevant options are $Q_{21}$ (Option 1) and $Q_{17}$ (Option 3).
• Consider the number associated with Q in these options: 21 or 17.
• Let's examine the sequence with 17: 3, 4, 7, 8, 9, 17, 11, 13, 24.
• For the pair $Q_{17}$, the letter Q is the 17th letter of the alphabet, and the number is 17. This match (alphabet position = number) is specific to this term.
• Although the overall number sequence (3, 4, 7, 8, 9, 17, 11, 13, 24) does not follow a simple arithmetic progression, the clear letter pattern combined with the specific property of the pair $Q_{17}$ (where the number matches the letter's alphabetical index) identifies it as the correct replacement for the question mark.

Conclusion

The missing term has the letter Q (based on the +3 pattern) and the number 17. Thus, the correct option is $Q_{17}$.