题目109：在飞镖游戏中一个玩家能够以多少种方式以100分以下离场？

欧拉计划

Darts

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.

The 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.

At 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.

There 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".

When 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).

There are exactly eleven distinct ways to checkout on a score of 6:

Note 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.

In addition we shall not include misses in considering combinations; for example, D3 is the same as 0 D3 and 0 0 D3.

Incredibly there are 42336 distinct ways of checking out in total.

How many distinct ways can a player checkout with a score less than 100?

题目：

在飞镖游戏中，一个玩家向标靶投掷三支飞镖。标靶被分成 20 个大小相等的区域，从 1 到 20 编号。

一支飞镖的得分取决于该飞镖落在的区域的编号。落在红/绿色区域外侧的飞镖得分为 0。环内的黑色和奶油色区域代表单倍得分。但是外侧的红/绿色环和中间的红/绿色环分别代表双倍和三倍得分。

在标靶的中心有两个同心环，叫做牛区，或者牛眼。外侧的牛眼是 25 分，内侧牛眼是双倍，也就是 50 分。

该游戏的规则有很多种，但在最流行的玩法中，玩家从 301 分或者 501 分开始，第一个将得分减为 0 的玩家获胜。但是，更普遍的玩法是一种“双倍结束”的系统，意思是玩家要赢得比赛必须将最后一支飞镖落在双倍得分区域（包括内侧牛眼）来结束比赛；如果在前两支飞镖时将玩家的得分减到 1 或者更少的话，则这一局飞镖游戏就算失败。

如果一个玩家能够以某个分数结束比赛的话，我们称其为一个“离场”。最高的离场是 170：T20 T20 D25（两个三倍 20 分加一个双倍牛眼）。

以 6 分开始一共有 11 中不同的离场方式：

注意 D1 D2 和 D2 D1 是不同的，因为它们以不同的方式结束。但是 S1 T1 D1 和 T1 S1 D1 被认为是相同的。

此外，我们不考虑投失的情况；例如 D3 和 0 D3 以及 0 0 D3 是相同的。

令人惊奇的是一共有 42336 中不同的离场方式。

一个玩家共有多少种初始分不超过 100 分的离场方式？

加餐

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