Cribbage is a card game that is played with a standard deck of \(52\) cards. There are four suits: clubs ♣, diamonds ♦️, hearts ♥️️, and spades ♠. Each suit has \(13\) cards identified with either a letter or number as follows: \(\text{A}\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\), \(10\), \(\text{J}\), \(\text{Q}\), \(\text{K}\). These letters and numbers are called the card’s rank. Since there are four suits, there are four cards with the same rank in every deck. Note: In this activity we will ignore the suits and just focus on the ranks.
To determine the number of points a hand is worth, each player makes combinations using \(2\), \(3\), \(4\), or \(5\) of their cards. Cards can be used more than once in different combinations. These are the combinations that count for points:
Each combination that totals \(15\) is worth \(2\) points. When making these combinations, an \(\text{A}\) is worth \(1\), a \(\text{J}\), \(\text{Q}\), or \(\text{K}\) is worth \(10\), and all number cards are worth their rank. For example:
\(6 + 9 = 15\) is worth \(2\) points
\(5 + \text{Q} = 15\) is worth \(2\) points
\(2 + 4 + 9 = 15\) is worth \(2\) points
Each pair of cards that have the same rank is worth \(2\) points.
Each combination of three or more cards that form a sequence of consecutive ranked cards is worth the number of cards in the sequence. For example:
\(3\), \(4\), \(5\) is worth \(3\) points
\(8\), \(9\), \(10\), \(\text{J}\) is worth \(4\) points
Which of the following hands is worth the most points?
Hand A: \(5\)♥️️, \(10\)♦️, \(\text{J}\)♠, \(\text{Q}\)♣, and \(\text{K}\)♠
Hand B: \(\text{A}\)♠, \(6\)♣, \(7\)♦️, \(7\)♥️️, and \(8\)♣
Hand C: \(5\)♣, \(5\)♥️️, \(5\)♠, \(8\)♠, and \(\text{Q}\)♦️
Justify your answer.
We record the combinations in each hand that earn points in the following tables.
| Type of Combination | Combinations | Total Points |
|---|---|---|
| Cards that total \(15\) | \(5\)♥️️ \(+\) \(10\)♦️, \(5\)♥️️ \(+\) \(\text{J}\)♠, \(5\)♥️️ \(+\) \(\text{Q}\)♣, \(5\)♥️️ \(+\) \(\text{K}\)♠ |
\(2+2+2+2 = 8\) |
| Pairs of cards with the same rank | none | \(0\) |
| Sequences of consecutive ranked cards | \(10\)♦️, \(\text{J}\)♠, \(\text{Q}\)♣, \(\text{K}\)♠ | \(4\) |
The total points in this hand is \(8+4=12\).
| Type of Combination | Combinations | Total Points |
|---|---|---|
| Cards that total \(15\) | \(7\)♦️ \(+\) \(8\)♣, \(7\)♥️️ \(+\) \(8\)♣, \(\text{A}\)♠ \(+\) \(6\)♣ \(+\) \(8\)♣, \(\text{A}\)♠ \(+\) \(7\)♦️ \(+\) \(7\)♥️️ |
\(2+2+2+2 = 8\) |
| Pairs of cards with the same rank | \(7\)♦️ and \(7\)♥️️ | \(2\) |
| Sequences of consecutive ranked cards | \(6\)♣, \(7\)♦️, \(8\)♣ \(6\)♣, \(7\)♥️️, \(8\)♣ |
\(3+3=6\) |
The total points in this hand is \(8+2+6=16\).
| Type of Combination | Combinations | Total Points |
|---|---|---|
| Cards that total \(15\) | \(5\)♣ \(+\) \(\text{Q}\)♦️, \(5\)♥️️ \(+\) \(\text{Q}\)♦️, \(5\)♠ \(+\) \(\text{Q}\)♦️, \(5\)♣ \(+\) \(5\)♥️️ \(+\) \(5\)♠ |
\(2+2+2+2 = 8\) |
| Pairs of cards with the same rank | \(5\)♣ and \(5\)♥️️, \(5\)♣ and \(5\)♠, \(5\)♥️️ and \(5\)♠ |
\(2+2+2=6\) |
| Sequences of consecutive ranked cards | none | \(0\) |
The total points in this hand is \(8+6=14\).
Therefore, Hand B is worth the most points.