Cut excess skin off drumsticks, if desired, then arrange chicken drumsticks and potatoes in a single layer on baking sheet.
Mix olive oil and seasonings in a separate small bowl, then brush chicken on both sides with olive oil mixture.
Drizzle potatoes with the rest of the seasoned olive oil and stir to coat.
Bake drumsticks and potatoes at 400 degrees for 50 minutes, flipping drumsticks over and stirring potatoes halfway through, until drumsticks are cooked through.
Serve your chicken and potatoes with a green vegetable or salad for a complete easy dinner.
Notes
I had large drumsticks here; if yours are smaller you may need to adjust the cooking time downward a little to compensate.