data/problems/109.yml
---
:id: 109
:name: Darts
:url: https://projecteuler.net/problem=109
:content: "In the game of darts a player throws three darts at a target board which
is split into twenty equal sized sections numbered one to twenty.\n\n  \n\nThe score of a dart is determined by the number of the region
that the dart lands in. A dart landing outside the red/green outer ring scores zero.
The black and cream regions inside this ring represent single scores. However, the
red/green outer ring and middle ring score double and treble scores respectively.\n\nAt
the centre of the board are two concentric circles called the bull region, or bulls-eye.
The outer bull is worth 25 points and the inner bull is a double, worth 50 points.\n\nThere
are many variations of rules but in the most popular game the players will begin
with a score 301 or 501 and the first player to reduce their running total to zero
is a winner. However, it is normal to play a \"doubles out\" system, which means
that the player must land a double (including the double bulls-eye at the centre
of the board) on their final dart to win; any other dart that would reduce their
running total to one or lower means the score for that set of three darts is \"bust\".\n\nWhen
a player is able to finish on their current score it is called a \"checkout\" and
the highest checkout is 170: T20 T20 D25 (two treble 20s and double bull).\n\nThere
are exactly eleven distinct ways to checkout on a score of 6:\n\n|  \nD3 |  \n |  \n |\n| D1 | D2 | |\n| S2 | D2 | |\n| D2 | D1
| |\n| S4 | D1 | |\n| S1 | S1 | D2 |\n| S1 | T1 | D1 |\n| S1 | S3
| D1 |\n| D1 | D1 | D1 |\n| D1 | S2 | D1 |\n| S2 | S2 | D1 |\n\nNote that D1 D2
is considered **different** to D2 D1 as they finish on different doubles. However,
the combination S1 T1 D1 is considered the **same** as T1 S1 D1.\n\nIn addition
we shall not include misses in considering combinations; for example, D3 is the
**same** as 0 D3 and 0 0 D3.\n\nIncredibly there are 42336 distinct ways of checking
out in total.\n\nHow many distinct ways can a player checkout with a score less
than 100?\n\n"