Trametinib or Combination Chemotherapy in Treating Patients With Refractory or Advanced Biliary or Gallbladder Cancer or That Cannot Be Removed by Surgery

Inclusion Criteria: * DISEASE RELATED CRITERIA * Patients must have histologically or cytologically documented carcinoma primary to the intra- or extra-hepatic biliary system or gall bladder with clinical and/or radiologic evidence of unresectable, locally advanced or metastatic disease; patients with ampullary carcinoma are not eligible * Patients must have measurable disease; computed tomography (CT) scans or magnetic resonance imaging (MRI)s used to assess measurable disease must have been completed within 28 days prior to registration; CT scans or MRIs used to assess non-measurable disease must have been completed within 42 days prior to registration; all disease must be assessed and documented on the Baseline Tumor Assessment Form (Response Evaluation Criteria in Solid Tumors \[RECIST\] 1.1) * PRIOR/CONCURRENT THERAPY CRITERIA * Patients must have completed any prior chemotherapy at least 21 days prior to registration and have recovered from any of the effects AND * Patients must have experienced progression to no more than 1 prior regimen of systemic chemotherapy for advanced biliary cancer OR * Patients who received adjuvant chemotherapy and had evidence of disease recurrence within 6 months of completion of the adjuvant treatment are also eligible; if patient received adjuvant treatment and had disease recurrence after 6 months, patients will only be eligible after failing one regimen of systemic chemotherapy used to treat the (unresectable or metastatic) diseas…