Puzzle #53: Celebration

At the time that I originally ran this puzzle, it was coming up on my fiftieth birthday. The student newspaper at my alma mater was promoting something for November 4 (Hugo’s Birthday). I do not know what that was about. My birthday is November 5. There was no official celebration, so I suggested we fake it:

Suppose you are in charge of the fireworks display. You have fifty charges of fireworks. You can create a tiny effect with one charge, a small effect with two, a medium with four, a large with six, and a real doozy with ten.

Your fireworks display must use exactly fifty charges. It must have at least one of each of the five sizes of effects, and for any two effects, there must be more of the smaller one. Given these constraints, answer:

1) How many possible selections of charges are there?
2) How many possible fireworks displays are there?
3) How many fireworks displays end with a real doozy?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, June 11, 2014> at noon Pacific Time. I will post the answer shortly after.