A Study to Evaluate the Safety, Tolerability, and Efficacy of the Addition of Omarigliptin (MK-3102) to Participants With Type 2 Diabetes Mellitus Who Have Inadequate Glycemic Control on Metformin Therapy (MK-3102-024)

Omarigliptin 25 mg capsule administered orally once weekly (preferably on the same day of each week)

Matching placebo to omarigliptin capsule administered orally once weekly (preferably on the same day of each week)

Glimepiride 1 or 2 mg tablet/capsule administered orally once daily and up-titrated to a maximum dose of 6 mg daily. Participants rescued with open-label glimepiride during Phase A will not receive glimepiride or matching placebo to glimepiride during Phase B.

Matching placebo to glimepiride tablet/capsule administered orally once daily and up-titrated to a mock maximum dose of 6 mg daily. Participants rescued with open-label glimepiride during Phase A will not receive glimepiride or matching placebo to glimepiride during Phase B.

During Phase B of the study, participants who received a maximum up-titration of open-label glimepiride or blinded glimepiride/matching placebo to glimepiride, may be rescued with open-label insulin glargine.

Participants continue stable pre-study dose of metformin tablet(s) administered orally (\>= 1500 mg daily) throughout the study.