• For Cohorts A, B, C, D, E, G, J: Has histologically- or cytologically-confirmed adenocarcinoma of the prostate without small cell histology

⁃ For Cohorts F, H, I: Has t-NE or de novo metastatic prostate cancer defined by ≥1% neuroendocrine cells that are located in discrete regions of a recent biopsy specimen from a metastasis as determined by the investigational site and confirmed by central histology review prior to enrollment. Epstein criteria of neuroendocrine differentiation in prostate cancer is used for eligibility. Specimens must have one of the morphologies of Small cell carcinoma or Large cell neuroendocrine carcinoma (LCNEC) or Mixed (small or large cell) NE carcinoma - acinar adenocarcinoma with positive IHC confirmed by central pathology review

• Is able to provide tumor tissue from a site not previously irradiated as follows: Cohorts A, E, G and J: must provide a core or excisional biopsy from soft tissue or bone biopsy within 1 year of screening and after developing mCRPC; Cohort B: must provide an archival tumor tissue sample or tumor tissue from a newly obtained core or excisional biopsy from soft tissue if the lesion is clinically accessible; Cohorts C and D with soft tissue disease: must provide a core or excisional biopsy from a soft tissue lesion if clinically accessible within 1 year of screening and after developing mCRPC and an archival specimen if available; and Cohorts F, H, and I must provide a core or excisional biopsy from soft tissue or a bone biopsy. For de novo metastatic neuroendocrine prostate participants, biopsies must be performed within 1 year of screening. Participants with bone metastasis only must provide an archival tumor tissue specimen

• Has prostate cancer progression within 6 months prior to screening, as determined by the investigator, by means of one of the following: PSA progression as defined by a minimum of 2 rising PSA levels with an interval of ≥1 week between each assessment where the PSA value at screening should be ≥2 ng/mL; radiographic disease progression in soft tissue based on Response Evaluation Criteria In Solid Tumors Version 1.1 criteria with or without PSA progression; radiographic disease progression in bone defined as the appearance of 2 or more new bone lesions on bone scan with or without PSA progression. Participants with de novo neuroendocrine prostate cancer will not need to provide evidence of progression within 6 months

• Has ongoing androgen deprivation with serum testosterone \<50 ng/dL (\<2.0 nM). Treatment with luteinizing hormone-releasing hormone agonists or antagonists for all cohorts must have been initiated ≥4 weeks prior to first dose of study therapy and must be continued throughout the study. Participants with de novo metastatic NE prostate cancer will not be required to have been previously treated with androgen deprivation therapy (ADT). ADT must be started in these participants by the time of treatment allocation/randomization

• Participants receiving bone resorptive therapy (including, but not limited to bisphosphonate or receptor activator of nuclear factor kappa-β ligand inhibitor) must be on stable doses for ≥4 weeks prior to first dose of study therapy

• Must be abstinent from heterosexual intercourse, refrain from donating sperm, or agree to use contraception (unless confirmed to be azoospermic) during the intervention period starting with the first dose of study therapy. The length of time required to continue contraception after the last dose of study intervention for each study intervention is as follows: 7 days for abiraterone acetate and lenvatinib; 30 days for enzalutamide; and 95 days for olaparib, docetaxel, and carboplatin/etoposide. No contraception measures are required during and after the intervention period for pembrolizumab/vibostolimab coformulation

• Has a performance status of 0, 1, or 2 on the Eastern Cooperative Oncology Group (ECOG) Performance Scale for Cohorts A and C and a performance status of 0 or 1 for Cohorts B, D, E, F, G, H, I and J within 10 days of study start

• For Cohort A: Has received docetaxel for mCRPC. Prior treatment with 1 other chemotherapy for mCRPC is allowed. Up to 2 second-generation hormonal manipulations (e.g., abiraterone acetate and/or enzalutamide) are allowed. Prior docetaxel for metastatic hormone-sensitive prostate cancer is allowed if ≥4 weeks have elapsed from the last dose of docetaxel prior to day 1 of Cycle 1

• For Cohort B: Has received prior treatment with either abiraterone acetate or enzalutamide (but not both) in the prechemotherapy mCRPC state. Participants in Cohort B must have received at least 4 weeks of either abiraterone or enzalutamide treatment (but not both) who failed treatment or became intolerant of the drug

• For Cohort C: Has received prior treatment with abiraterone acetate in the pre-chemotherapy mCRPC state without prior enzalutamide. Participants in Cohort C must have received at least 4 weeks of abiraterone treatment who failed treatment or become intolerant of the drug. Participants who received abiraterone acetate in the hormone-sensitive state will not be eligible

• For Cohort D: Has not received chemotherapy for mCRPC and has either not had prior second generation hormonal manipulation for mCRPC OR has previously been treated with enzalutamide for mCRPC and failed treatment or has become intolerant of the drug. Prior docetaxel for metastatic hormone-sensitive prostate cancer is allowed if ≥4 weeks have elapsed from the last dose of docetaxel. Prior treatment with abiraterone acetate in the hormone-sensitive metastatic setting is allowed as long as there was no progression on this agent and abiraterone acetate was not discontinued due to adverse events (AEs)

• For Cohorts E, G and J: Has received docetaxel for mCRPC. Prior treatment with 1 other chemotherapy for mCRPC is allowed. Up to 2 second-generation hormonal manipulations (eg, abiraterone acetate, enzalutamide, apalutamide, darolutamide or other next-generation hormonal agents \[NHA\]) are allowed. Participants who received prior ketoconazole for metastatic disease may be enrolled. If docetaxel chemotherapy is used more than once (eg, once for metastatic hormone-sensitive and once for mCRPC), it will be considered as 1 therapy. Prior docetaxel for metastatic hormone-sensitive prostate cancer (mHSPC) is allowed if ≥4 weeks have elapsed from the last dose of docetaxel prior to Day 1 of Cycle 1

• For Cohort F, H, and I: Participants must have received prior treatment with androgen deprivation therapy (ADT) for metastatic disease. Prior treatment with up to a total of 2 chemotherapies for mCRPC is allowed, as well as up to 2 second-generation hormonal manipulations for mCRPC. Participants who received prior ketoconazole for metastatic disease may be enrolled. Docetaxel for mHSPC is allowed in addition to docetaxel for mCRPC. If docetaxel chemotherapy is used more than once (eg, once for metastatic hormone-sensitive and once for mCRPC), it will be considered as 1 therapy