If you used only photons that were dropping from the n = 6 shell to the n = 1, how many of these photons would be required to keep a 100 Watt light bulb lit for 10 minutes?

Answer 1

You don't have enough information to answer this. I assume you are talking about the electron in a H atom. It will different for other elements.
How much energy do you get from a photon falling from n = 6 to n = 1? That's delta E = 2.180E-18J(1/1^2 - 1/6^2) = ?J/photon that you have.

How much energy is required to light a 100 watt bulb for 10 minutes?
100 watts = 100 J/s and for 10 minutes is 100 J/s x 10 min x (60 s/min)= ?J needed.
?J/photon x #photons = ?J needed.
Solve for # photons.