Step 1: Understanding Syphilis. Syphilis is a sexually transmitted infection (STI) caused by the bacterium \( \text{Treponema pallidum} \). It is a spirochete bacterium that primarily spreads through sexual contact or from mother to fetus during pregnancy (congenital syphilis).
Step 2: Comparison with other organisms. - Option \( (A) \): \( \text{Vibrio cholerae} \) causes cholera, an acute diarrheal disease.
- Option \( (C) \): \( \text{Bacillus pertussis} \) causes whooping cough (pertussis).
- Option \( (D) \): \( \text{Clostridium tetani} \) causes tetanus, characterized by severe muscle spasms.
Step 3: Conclusion. Only \( \text{Treponema pallidum} \) is the causative organism for syphilis.
Conclusion: The correct answer is \( (B) \ \text{Treponema pallidum} \).