A Study of Reduced Radiation Therapy With Chemotherapy in People With HPV-Positive Throat Cancer

Inclusion Criteria: * Pathologically (histologically or cytologically) proven diagnosis of HPV associated squamous cell carcinoma of the oropharynx (tonsil, base of tongue, or oropharyngeal walls) from biopsy, surgical resection or excisional biopsy regardless of margin status. 1. Squamous cell carcinoma of the neck of unknown primary is allowed with excision biopsy of a lymph node (or core biopsy) or consent from the PI or co-PI. Patient must have excisional biopsy or core biopsy done in order to be on protocol. Note: Evidence of HPV associated oropharyngeal cancer from either the primary tumor site or from a lymph node. A patient is HPV positive when he or she tests positive having tested positive for both p16 expression (70% nuclear and cytoplasm expression; Ventana Medical Systems) and mRNA HPV in situ hybridization. * Subjects must have clinically or radiographically evident measurable gross disease at either the primary tumor site or nodal stations. * Oropharyngeal Carcinoma (AJCC, 7th ed.) without evidence of distant metastasis based on FDG PET/CT. * CT or MRI of the neck with and without contrast Note: A CT scan of neck and/or a PET/CT performed for the purposes of radiation planning may serve as planning tools. * ECOG Performance Status of 0-2 or KPS ≥ 70 * Age ≥ 18 * Adequate hematologic function within 30 days prior to registration, defined as follows: 1. White Blood Count (WBC) ≥ 2 K/mcL 2. Absolute neutrophil count (ANC) ≥ 1,000 cells/mm\^3 3. Platelets ≥ 100,000 cells/mm\^3 4. Hemoglobin ≥ 8.0 g/dl Note: The use of transfusion or other intervention to achieve Hgb ≥ 8.0 g/dl is acceptable * Adequate renal function within 30 days prior to registration, defined as follows: a. Serum creatinine ≤ 1.5 mg/dl or creatinine clearance (CC) ≥ 50 ml/min determined by 24-hour collection or estimated by Cockcroft-Gault formula CCr male = \[(140 - age) x (wt in kg)\] \[(Serum Cr mg/dl) x (72)\] CCr female = 0.85 x (CrCl male) Note: Patients who cannot tolerate cisplatin or carboplatin/5FU based on clinical judgment will receive carboplatin and paclitaxel * Adequate hepatic function within 30 days prior to registration, defined as follows: 1. Bilirubin ≤ 2 mg/dl 2. AST or ALT ≤ 3 x the upper limit of normal Note: Patients who cannot tolerate cisplatin or carboplatin/5FU based on clinical judgment will receive carboplatin and paclitaxel. Negative serum pregnancy test within 14 days prior to registration for women of childbearing potential * The subject must provide study-specific informed consent prior to study entry Subject able to undergo MRI scans except for major medical contraindications like presence of a pacemaker or approved by the PI or the CO-PI that the subject does not need to undergo MRI scans Exclusion Criteria: * Subjects with prior head and neck radiation therapy * Subjects with simultaneous primary cancers outside of the oropharynx if determined by the PI/Co-PI the patient can proceed with protocol activities. Note: Exceptions can be made for patients with simultaneous primaries outside the oropharynx if determined by the PI/Co-PI the patient can proceed with protocol activities. * Prior invasive malignancy (except non-melanomatous skin cancer) unless disease free for 3 years or if cure rate from treatment at 5 years to be 90% or greater * Prior systemic chemotherapy for the study cancer; note that prior chemotherapy for a different cancer is allowable * Severe, active co-morbidity defined as follows (exceptions can be made if approved by the PI and/or co-PI) 1. Unstable angina and/or congestive heart failure requiring hospitalization within the last 6 months 2. Transmural myocardial infarction within the last 6 months 3. Acute bacterial or fungal infection requiring intravenous antibiotics at the time of registration 4. Chronic Obstructive Pulmonary Disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy within 30 days of registration 5. Hepatic Insufficiency resulting in clinical jaundice and/or coagulation defects