I know this question has been asked many time but my question is bit different here. What i mean by the above question is that I recently got to learn that light actually does not exclusively take the path taking least time. Light, infact, also takes nearby path other than that taking the least time but the reason why we see light taking least time is that the other path interfere with one another and cancel each other out. Now my question is that exactly how does this interference and cancellation occur? Also why does this interference and cancellation also not occur with path taking the least time? I am not very advanced in math so please if anyone can try to theoretically explain this or try to go easy on the math, I woud greatly appreciate it. Thanks.
Fermat's principle is one expression of it, you can see it derived at https://en.m.wikipedia.org/wiki/Fermat%27s_principle
It is a consequence of Huygens's principle, which is explained and where you see the wavelets cancelling out in the article atwikipedia at https://en.m.wikipedia.org/wiki/Huygens–Fresnel_principle. It is also derived there. It shows that rectilinear or plane wave propagation (i.e., the simplest mathematical way to figure it out from A to B, taking A and B at infinity) is also explained that way. Fresnel showed that in 1816.
The Feynman QM formulation is in his Lectures in Physics, you can google more. It has also been generalized in field and quantum field theory to the principle of least action, actually for any field, not just light. See for instance https://en.m.wikipedia.org/wiki/Principle_of_least_action