PRIMARY OBJECTIVES: I. To determine radiographic progression free survival with 4 cycles of docetaxel with carboplatin followed by maintenance rucaparib camsylate (rucaparib) in the treatment of patients with metastatic castration resistant prostate cancer with homologous recombination DNA repair deficiency. SECONDARY OBJECTIVES: I. To assess maximal prostate-specific antigen (PSA) response to induction docetaxel and carboplatin. II. To assess PSA response to switch maintenance with rucaparib. III. To assess PSA response duration to docetaxel and carboplatin followed by switch maintenance with rucaparib. IV. To assess response of measurable disease. OUTLINE: INDUCTION: Patients receive docetaxel intravenously (IV) and carboplatin IV on day 1. Treatment repeats every 21 days for 4 courses in the absence of disease progression or unacceptable toxicity. MAINTENANCE: Patients receive rucaparib camsylate orally (PO) twice daily (BID) on days 1-28. Courses repeat every 28 days in the absence of disease progression or unacceptable toxicity. After completion of study treatment, patients are followed up periodically.