Problem 928

Cribbage

This problem is based on (but not identical to) the scoring for the card game Cribbage.

Consider a normal pack of $52$ cards. A Hand is a selection of one or more of these cards.

For each Hand the Hand score is the sum of the values of the cards in the Hand where the value of Aces is $1$ and the value of court cards (Jack, Queen, King) is $10$.

The Cribbage score is obtained for a Hand by adding together the scores for:

Pairs. A pair is two cards of the same rank. Every pair is worth $2$ points.
Runs. A run is a set of at least $3$ cards whose ranks are consecutive, e.g. $9$, $10$, Jack. Note that Ace is never high, so Queen, King, Ace is not a valid run. The number of points for each run is the size of the run. All locally maximum runs are counted. For example, $2$, $3$, $4$, $5$, $7$, $8$, $9$, the two runs of $2$, $3$, $4$, $5$ and $7$, $8$, $9$ are counted but not $2$, $3$, $4$ or $3$, $4$, $5$.
Fifteens. A fifteen is a combination of cards that has value adding to $15$. Every fifteen is worth $2$ points. For this purpose the value of the cards is the same as in the Hand Score.

For example, $(5 \spadesuit, 5 \clubsuit, 5 \diamondsuit, K \heartsuit)$ has a Cribbage score of $14$ as there are four ways that fifteen can be made and also three pairs can be made.

The example $( A \diamondsuit, A \heartsuit, 2 \clubsuit, 3 \heartsuit, 4 \clubsuit, 5 \spadesuit)$ has a Cribbage score of $16$: two runs of five worth $10$ points, two ways of getting fifteen worth $4$ points and one pair worth $2$ points. In this example the Hand score is equal to the Cribbage score.

Find the number of Hands in a normal pack of cards where the Hand score is equal to the Cribbage score.

克里比奇纸牌

本题基于纸牌游戏克里比奇纸牌的计分规则（但不完全相同）。

考虑一副标准的$52$张牌扑克；一手牌是指从中选择的一张或多张牌。

对于每一手牌，其手牌得分是其牌面点数的总和，其中$A$的点数为$1$，花牌（$J$、$Q$、$K$）的点数为$10$。

另一方面，一手牌的克里比奇得分则是将以下得分相加：

对子：对子是两张相同点数的牌，每个对子值$2$分。
顺子：顺子是至少$3$张连续点数的牌，例如$9$、$10$、$J$。注意$A$被视为最小的牌，所以$Q$、$K$、$A$不是有效的顺子。每个顺子的得分等于顺子的长度，且计算得分时只考虑最长的顺子。例如，在$2$、$3$、$4$、$5$、$7$、$8$、$9$中，有$2$、$3$、$4$、$5$和$7$、$8$、$9$两个顺子，而不考虑$2$、$3$、$4$或$3$、$4$、$5$。
十五：十五是牌面点数之和为$15$的任意组合，每个十五值$2$分。这里的牌面点数与手牌得分中的牌面点数相同。

例如，$(5 \spadesuit, 5 \clubsuit, 5 \diamondsuit, K \heartsuit)$的克里比奇得分为$14$，因为这手牌可以组成四种十五和三种对子。

再例如，$( A \diamondsuit, A \heartsuit, 2 \clubsuit, 3 \heartsuit, 4 \clubsuit, 5 \spadesuit)$的克里比奇得分为$16$：两种五张的顺子$10$分，两种十五$4$分，一种对子$2$分。在这个例子中，手牌得分恰好等于克里比奇得分。

在一副标准扑克中，求手牌得分恰好等于克里比奇得分的手牌数目。