Calculating 5-Card Hands Containing Exactly 4 Cards from One Suit

When dealing with a standard deck of cards, one common problem is determining the number of ways to select a hand that includes exactly four cards from any one suit. This question often arises in the study of combinatorics and poker probability.

Understanding the Problem

To solve this problem, we can break it down into several cases and use combinatorial mathematics to find the total number of possible hands. A standard deck has 52 cards, with 13 cards in each of the 4 suits: diamonds, clubs, hearts, and spades. The question is asking for the number of 5-card hands that include exactly four cards from any one suit and a fifth card from a different suit.

Solution Breakdown

There are four cases to consider, depending on which suit has the four cards. We will calculate the number of possible hands for one of these cases and then multiply the result by four, as the other cases are symmetrical.

Case Analysis

Case A: Four Hearts and One Non-Heart

First, we calculate the number of ways to choose four hearts from the 13 available hearts. This can be done using the combination formula C(13, 4), which represents the number of ways to choose 4 items from 13 without regard to order. The formula for combinations is:

C(n, k) n! / (k!(n-k)!)

Applying this, we get:

C(13, 4) 13! / (4!(13-4)!) 13! / (4!9!) 715

Next, we need to choose one card from the remaining 39 cards (since the other 3 cards in the hand must come from a different suit). This is a simple combination of 39 cards taken 1 at a time, or C(39, 1) 39.

The total number of hands for this case is then the product of these two combinations:

715 * 39 27885

Since the problem is symmetric for all four suits, we multiply this result by 4 to account for all possible cases:

27885 * 4 111540

Verification and Generalization

To verify our solution, we can use the combinatorial notation and formulas provided. The general formula for the number of 5-card hands with exactly four cards from one suit is:

4 * C(13, 4) * C(39, 1)

Substituting the values, we get:

4 * 715 * 39 111540

This confirms our earlier calculation.

Conclusion

The number of 5-card hands that can be dealt that include exactly four cards from any one suit is 111,540. This result is derived by considering the combinatorial possibilities for each case and then summing these possibilities across all four suits.

References

Reference:

For further details and to verify calculations, you can use mathematical software such as PariGP. Here is the corresponding code:

pari:20:22 gp binomial(4, 1) * binomial(13, 4) * binomial(39, 1)

This code will yield the result 111540, confirming the solution.