Problem number 109

I love puzzles with pictures:

In a darts game, a player throws 3 arrows at a target, divided into twenty sections of the same size, numbered from one to twenty.


')

The points of each dart are determined by the number of the section in which it landed. If the dart landed abroad outside the outer red-green circle, then he scored zero. Black and beige areas inside the circle give a single number of points. Red-green outer and inner circles double and triple, respectively.

In the center of the target are two circles, called bulls-eye. The outer one gives 25, the inner doubling is 25, that is 50.

There are many options for the game, but the most popular is when players start with a 301 or 501 account and the first player who reduces his points to 0 wins. However, they usually play with a “double exit” system, which means that the player must hit the sector with a double with the last dart (including a double bulls-eye in the center of the target); any other hit that reduces the points to one or less means “brute force”.

The number of points at which a player can finish with a series of three shots is called “checkout”, with a maximum checkout of 170: T20 T20 D25 (two tripled 20ki and double bulls-eye).

There are exactly eleven ways to win with 6 points 6:

D3
D1	D2
S2	D2
D2	D1
S4	D1
S1	S1	D2
S1	T1	D1
S1	S3	D1
D1	D1	D1
D1	S2	D1
S2	S2	D1

Notice that D1 D2 is considered different from D2 D1, since they end in different doublings. However, the combination of S1 T1 D1 is identical to T1 S1 D1.

Additionally, we prefer not to include slips in the combination; for example, D3 is identical to 0 D3 and 0 0 D3.

There are 42336 different ways to finish.

And how many ways are there to finish with less than 100 points?


I consider solutions to publish and discuss incorrect ... There is a special forum on projecteuler.net

PS: It was necessary to do the topics closed, now everything is mined.