Carboplatin or Olaparib for BRcA Deficient Prostate Cancer

Inclusion Criteria: 1. Signed study informed consent form (ICF) and HIPAA authorization form 2. Diagnosis of prostate cancer (pure small-cell histology or pure high-grade neuroendocrine histology are excluded; neuroendocrine differentiation is allowed) 3. Ongoing gonadal androgen deprivation therapy with gonadotropin-releasing hormone (GnRH) analogues, antagonists or orchiectomy. Patients who have not had an orchiectomy must be maintained on effective GnRH analogue/antagonist therapy 4. mCRPC as defined by serum testosterone \< 50 nanograms per milliliter (ng/ml) for patients on GnRH analogues or antagonists) and at least one of the following: * Prostate-specific antigen (PSA) level of at least 2 nanograms per milliliter (ng/ml) that has risen on at least 2 successive occasions at least 1 week apart * Evaluable disease progression by modified RECIST 1.1 (Response Evaluation Criteria in Solid Tumors) * Progression of metastatic bone disease on bone scan, CT or MRI with \> 2 new lesions 5. Prior therapy with abiraterone acetate, enzalutamide, apalutamide, or darolutamide 6. Eastern Cooperative Oncology Group (ECOG) Performance Status of \< 2 (see Appendix 3, ECOG Grading Scale) 7. Results of previous standard DNA testing, or previous research testing, which confirms RAD51B, RAD51C, RAD51D, or RAD54L mutations (see Introduction, Section 2 for study design and previous research on targeted therapy) from primary, metastatic tumor or circulating tumor DNA, or pathogenic/likely pathogenic germline variant as assessed by a CLIA certified laboratory level assay for DNA sequencing. 8. Patients must have normal organ and bone marrow function measured within 28 days prior to administration of study treatment as defined below: * Hemoglobin \> 10.0 grams per deciliter (g/dL) * Absolute neutrophil count (ANC) \> 1.5 x 109/liter (L) * Platelet count \> 100 x 109/L * Total bilirubin \< 1.5 x institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) (Serum Glutamic Oxaloacetic Transaminase (SGOT) / Alanine aminotransferase (ALT) (Serum Glutamic Pyruvate Transaminase (SGPT) \< 2.5 x institutional upper limit of normal unless liver metastases are present in which case, they must be \< 5x ULN * Patients must have creatinine clearance estimated using the Cockcroft-Gault equation of \>51 mL/min: Estimated creatinine clearance =(140-age \[years\]) x weight (kg)/(serum creatinine (mg/dL) x 72) Exclusion Criteria: 1. Currently receiving active therapy for other neoplastic disorder(s) 2. Concurrent enrollment in another clinical investigational drug or device study 3. Histologic evidence of small cell carcinoma (morphology alone - immunohistochemical evidence of neuroendocrine differentiation without morphologic evidence is not exclusionary) 4. Prior treatment with platinum, mitoxantrone or PARP inhibitor for castration resistant prostate cancer 5. Known parenchymal brain metastasis 6. Active or symptomatic viral hepatitis or chronic liver disease AST or ALT \> 2.5 x ULN or total bilirubin \> ULN (unless Gilbert's syndrome is the etiology of hyperbilirubinemia) 7. Subjects with myelodysplastic syndrome/acute myeloid leukemia or with features suggestive of MDS/AML 8. Concomitant use of known strong CYP3A inhibitors (e.g. itraconazole, telithromycin, clarithromycin, protease inhibitors boosted with ritonavir or cobicistat, indinavir, saquinavir, nelfinavir, boceprevir, telaprevir) or moderate CYP3A inhibitors (e.g. ciprofloxacin, erythromycin, diltiazem, fluconazole, verapamil). The required washout period prior to starting olaparib is 2 weeks 9. Concomitant use of known strong (e.g. phenobarbital, enzalutamide, phenytoin, rifampicin, rifabutin, rifapentine, carbamazepine, nevirapine and St John's Wort) or moderate CYP3A inducers (e.g. bosentan, efavirenz, modafinil). The required washout period prior to starting olaparib is 5 weeks for phenobarbital and 3 weeks for other agents 10. Subjects unable to swallow orally administered medication and subjects with gastrointestinal disorders likely to interfere with absorption of the study medication 11. Clinically significant heart disease as evidenced by myocardial infarction, or arterial thrombotic events in the past 6 months, severe or unstable angina, or New York Heart Association (NYHA) Class II-IV heart disease or cardiac ejection fraction measurement of \< 35 % at baseline 12. Treatment with an investigational therapeutic within 30 days of Cycle-1 13. Presence of dementia, psychiatric illness, and/or social situations limiting compliance with study requirements or understanding HIPAA authorization and/or giving of informed consent 14. Any condition(s), medical or otherwise, which, in the opinion of the Investigators, would jeopardize either the patient or the integrity of the data obtained