Calculating 5-Card Combinations in a Standard Deck with At Least One King

This article explores how to calculate the number of 5-card combinations from a deck of 52 cards, given that each selection of 5 cards must contain at least one king. We'll break down the problem into smaller, manageable parts and provide detailed calculations to ensure clarity.

Introduction to 5-Card Combinations

In the world of card games, combinations play a crucial role in determining various probabilities. Specifically, we are interested in combinations where any given 5-card hand from a standard deck (52 cards) must contain at least one king. There are four kings in a standard deck, and the other 48 cards are non-kings. This article will guide you through the steps required to determine how many such combinations exist.

Calculation of Combinations

The total number of 5-card combinations from a deck of 52 cards can be expressed using the combination formula:

nCr n! / (r!(n-r)!)

Applying this formula for a 52 card deck, where n 52 and r 5:

52C5 2598960

Combinations without Any Kings

Next, we need to calculate the number of 5-card combinations that do not contain any kings. Since we are removing all 4 kings from the deck, we are left with 48 non-king cards. The number of ways to choose 5 cards from 48 non-king cards is:

48C5 1712304

Combinations with at Least One King

To find the number of 5-card combinations with at least one king, we subtract the number of combinations without any kings from the total number of combinations:

Total 5-card combinations with at least one king 52C5 - 48C5

Substituting the values:

Total 5-card combinations with at least one king 2598960 - 1712304 886656

Detailed Breakdown of Combinations

We can also break down the process in a more detailed manner to illustrate each step:

Step 1: Exactly 1 King

If a 5-card hand contains exactly one king, we need to choose 1 king from the 4 available kings and then choose 4 non-kings from the 48 non-kings.

4C1 * 48C4 4 * 194580 778320

Step 2: Exactly 2 Kings

If a 5-card hand contains exactly two kings, we need to choose 2 kings from the 4 available kings and then choose 3 non-kings from the 48 non-kings.

4C2 * 48C3 6 * 17296 103776

Step 3: Exactly 3 Kings

If a 5-card hand contains exactly three kings, we need to choose 3 kings from the 4 available kings and then choose 2 non-kings from the 48 non-kings.

4C3 * 48C2 4 * 1128 4512

Step 4: Exactly 4 Kings

Finally, if a 5-card hand contains exactly four kings, we need to choose 4 kings from the 4 available kings and then choose 1 non-king from the 48 non-kings.

4C4 * 48C1 1 * 48 48

Total Number of Combinations

Adding all these possibilities together:

778320 103776 4512 48 886656

Thus, there are 886,656 possible 5-card combinations from a standard deck where each selection of 5 cards has at least one king.