Interesting question!
"Best of luck" seems to say, you hope that someone will have the best luck they possibly can with a specific something (or everything in general), such that if there's any part of their experience that's left up to chance, it will go well for them, because they have "the best luck."
"All the best," which I often see as "I wish you all the best," seems to say, you hope that a specific something (or everything) will happen in the best way possible for them, regardless of how it ended up that way.
For example, wishing "all the best" means you don't know whether she'll do well because she studied hard or because the questions were ridiculously easy or because the questions happened to be about the few things she did know from the material she wasn't able to study; you just want her to do well and wish that she will.
On the other hand, wishing "best of luck" is only wishing her good luck, so only applies to situations of random chance. In this case, the questions from Ch. 10 lining up with her potentially low amount of knowledge from the chapter would be luck, and some would say that the test being easy in general would also be luck.
A potential source of confusion: "Make the best of (whatever)" is a very different phrase from "best of luck". To "make the best of x" means that regardless of how bad situation x turned out because of chance, the subject will make every effort to get a good result out of it anyway. This phrase tends to connote that x was something that did have an element of chance and that it went poorly.
So, if I heard someone say, "We'll make the best of our luck," I would assume that they ended up with circumstances that they didn't want, which are either bad or maybe simply strange, but they are determined to utilize what they do have as well as they possibly can.
Hope it helps!
It's let the world know how you feel.
It's just because let is followed by the base form of the verb. (AKA infinitive without to.)
Let him know,
Let them know,
Let him/them knows.
Best Answer
"Let me know" is asking to be informed. As "let" means to allow, this is a polite way of asking to be kept informed, because you are asking for permission to know.
"Tell me" is a demand, and so less polite. You can certainly make it more polite by saying "please tell me", but you are still asking for information rather than asking to be allowed that information. "Please let me know" would be even more polite.