一款名为“快乐家庭”的卡牌游戏共有 28 张牌，由 7 套卡牌组成，每套 4 张。每套卡牌包括来自同一个家庭的父亲、母亲、儿子和女儿。家庭名称分别为：画家、邮递员、水管工、屠夫、木匠、歌手和教师。例如，“教师”家庭的完整整套卡牌包括教师父亲、教师母亲、教师儿子和教师女儿。 游戏的目标是收集尽可能多的完整家庭套牌。在游戏结束时，所有卡牌都会被收集。每个玩家要么没有剩余卡牌，要么拥有完整的家庭套牌。拥有最多完整家庭套牌的玩家即为赢家。 A 和 B 玩一局“快乐家庭”游戏。

1. Determine the number of sets B must have and the available pool of sets to choose from.

Since there are 7 complete sets in total and A has exactly 3 sets, B must hold the remaining $7 - 3 = 4$ sets.
A has the Teacher set, so B cannot have it. Thus, B's 4 sets must be formed from the remaining $7 - 1 = 6$ sets.

2. Calculate the number of combinations for B.

Number of combinations = $^6C_4 = 15$

1. Determine the number of complete sets A must have in order to win.

Since there are $7$ complete sets in total and A is the winner, A must have strictly more sets than B.
Therefore, A must have $4$, $5$, $6$, or $7$ complete sets.

2. Calculate the total number of combinations of complete sets A can have.

Total possible combinations = $^7C_4 + {}^7C_5 + {}^7C_6 + {}^7C_7$
$= 35 + 21 + 7 + 1$
$= 64$

1. Find the number of ways to arrange the non-mother cards in a circle.

There are 3 sets of cards, meaning there are 3 mothers and $12 - 3 = 9$ non-mother cards.

Number of ways to arrange the 9 distinct non-mother cards in a circle $= (9-1)! = 8! = 40\,320$.

2. Find the number of ways to arrange the 3 mother cards such that no two are together.

Arranging the 9 non-mother cards creates exactly 9 spaces between them.
To ensure no two mothers are next to each other, we insert the 3 distinct mother cards into these 9 spaces.

Number of ways to choose spaces and arrange the mothers $= ^9P_3 = 9 \times 8 \times 7 = 504$.

3. Calculate the total number of arrangements.

Total number of arrangements $= 8! \times ^9P_3 = 40\,320 \times 504 = 20\,321\,280$.