Inclusion Criteria:

        All Treatment Arms:

          1. Male or female participants 18 years or older.

          2. Participants who, in the opinion of the treating physician, have failed standard
             therapies and for whom a phase 1 trial is an appropriate option.

          3. Radiographically or clinically evaluable tumor. For expansion phase: Tumors must be
             measurable and of the protocol specified genetic mutational status, where applicable.

          4. Recovered (ie, less than or equal to [<=] Grade 1 toxicity) from adverse effects
             (except alopecia) of prior therapy.

          5. Eastern Cooperative Oncology Group (ECOG) performance status 0-1.

          6. Expected survival time of at least 3 months in the opinion of the investigator.

          7. Block of banked tumor tissue and/or greater than or equal to (>=) 10 unstained slides.
             Participants who satisfy all other eligibility criteria but do not have banked
             tissue/slides may be asked to consent to baseline biopsy.

          8. Suitable vein access for the study-required blood sampling.

          9. Thyroid function tests consistent with stable thyroid function. Note: Participants on
             a stable dose of thyroid replacement therapy for a suggested minimum of 12 weeks
             before Cycle 1, Day 1 are eligible.

         10. Left ventricular ejection fraction (LVEF) of 50 percent (%) or greater, as measured by
             echocardiogram (ECHO) or multiple-gated acquisition scan (MUGA), within 28 days before
             the first dose of MLN2480

         11. Female participants who are post-menopausal for at least 1 year, surgically sterile,
             or agree to practice 2 effective methods of contraception through 120 days (4 months)
             after the last dose of study drug for participants in Arms 1, 2, and 5, and through 6
             months for participants in Arms 3 and 4, or agree to practice true abstinence.

         12. Male participants who, even if surgically sterilized, agree to practice effective
             barrier contraception through 120 days (4 months) after the last dose of study drug
             for participants in Arms 1, 2, and 5, and through 6 months for participants in Arms 3,
             and 4, or agree to practice true abstinence.

         13. Additional inclusion criteria for arm 3 expansion only (MLN2480 + paclitaxel):

             a. Participants with Kirsten rat sarcoma viral oncogene homolog (KRAS) exon 2 or BRAF
             non-V600 mutation-positive non-small cell lung cancer (NSCLC) who have received a
             minimum of 1 but not more than 2 prior cytotoxic-approved regimens.

         14. Additional inclusion criteria for arms 4 and 5 expansion only (MLN2480 + cetuximab;
             MLN2480 + irinotecan):

               1. Participants with CRC who have received a minimum of 1 but not more than 2 prior
                  cytotoxic-approved regimens.

        Exclusion Criteria:

        All treatment arms:

          1. Female participants who are pregnant or currently breastfeeding.

          2. History of any serious medical or psychiatric illness that could, in the
             investigator's opinion, potentially interfere with safe protocol completion.

          3. History of uncontrolled brain metastasis unless: previously treated with surgery,
             whole-brain radiation, or stereotactic radiosurgery; stable disease for >= 60 days
             without steroid use (or stable steroid dose established for >= 28 days before the
             first dose of MLN2480).

          4. Ongoing seizure disorder or a requirement for antiepileptics.

          5. Recent prior therapies, including: chemotherapy and hormonal therapy <= 4 weeks or 4
             half lives, whichever occurs first, before administration of study drug;
             immunotherapy/monoclonal antibody use <= 4 weeks before administration of MLN2480; or
             radiation therapy <= 3 weeks before administration of study drug.

          6. Chronic therapeutic corticosteroid use with the exception of replacement therapy for
             adrenal insufficiency or corticosteroid inhalers.

          7. Known history of human immunodeficiency virus infection, hepatitis B, or hepatitis C;
             Prior allogeneic bone marrow or organ transplantation, or active condition of chronic
             immune suppression is not allowed.

          8. Concomitant use, or administration <= 14 days before first dose of study drug(s), of
             clinically significant enzyme inducers.

          9. Treatment with gemfibrozil (strong Cytochrome P4502C8 [CYP2C8] inhibitor) within 14
             days before the first dose of MLN2480.

         10. History of or current illicit drug use, drug abuse, or alcohol abuse.

         11. Major surgery within 14 days before the first dose of study drug.

         12. Inability to comply with study requirements.

         13. Other unspecified reasons that, in the opinion of the investigator or Millennium, make
             the participant unsuitable for enrollment.

         14. Additional exclusion criteria for arms 3, 5, and 6 expansion only (MLN2480 +
             paclitaxel; MLN2480 + irinotecan; MLN2480 monotherapy):

             a. Prior treatment with rapidly accelerated fibrosarcoma (RAF), extracellular
             signal-regulated kinases (MEK), or other inhibitors of the mitogen-activated protein
             kinase (MAPK) pathway.

         15. Additional exclusion criteria for arm 2 only (MLN2480 + alisertib):

             a. History of uncontrolled sleep apnea syndrome and other conditions that could result
             in excessive daytime sleepiness, such as severe chronic obstructive pulmonary disease.

         16. Additional exclusion criteria for arm 3 only (MLN2480 + paclitaxel):

             a. Known hypersensitivity to paclitaxel, or its components or other drugs formulated
             in Cremophor® EL (polyoxyethylated castor oil).

         17. Additional exclusion criteria for arm 5 only (MLN2480 + irinotecan):

               1. Use of strong or moderate Cytochrome P4503A (CYP3A) inhibitors <= days of the
                  first dose of irinotecan.