Math puzzles!

The chance that all twenty car plates will be different is:
100!/((100 - 20)! * (100^20)) = 13,04%

Or the complementary, the chance that at least one plate will be the same is 86,96%

So the odds are extremely against the guy betting they'll all be different (even a 1:5 bet is disadvantageous).
Not gonna calculate the expected value here, it's not hard though.

For a group of N possibilities, the chance that in n different picks all will be different is:

N!/((N - n)! * N^n))

Good good. Since you have the percentages already... the expected value is -1*86.96 + 1*13.04=-73.92 cents, if the pay out is 1:1. even at 5:1 the pay out is -1*86.96 + 5*13.04=-21.76... you'd need 7:1 payout for the game to be worth it for you.


Time for some darts trivia/problems


For reference, in darts, you throw darts at a circular board sectioned off into 20 pieces... think of cutting cake into 20 slices. The numbers 1 through 20 are mixed through the "slices" and landing there gives you that many points. There are also small sections which double or triple the score. So it's possible to get triple 11, for 33 points in one shot. The very center is a circle with another circle around it. The very center is worth 50 points, and the small ring around it is 25 points.




The goal in the particular game I'll ask about is to start off at a score (say 300) and work your way down to exactly zero. If you hit a 20, your score drops from 300 to 280. If you're at 5 and hit a 20, you stay at 5 (no going into negatives).

7) What's the lowest score that cannot be scored with one dart throw?
8) What is the lowest score that you can be left with where the game CANNOT be finished in two darts?
9) What is the lowest score that you can be left with where the game CANNOT be finished in THREE darts?
10) Is it possible to score 100 points using three triples? what about four triples?
11) There are four ways to score 26 points using a triple then a double... name them. Ex: triple 1 and double 6 gets you 15. But you must find 4 ways to get a 26 this way.



Try and answer only 1 question at a time to let others have a shot at solving 'em. Also, number your response please