To solve the problem of finding how many 5-card combinations can be made from a deck of 52 cards, where exactly 1 of the cards is a king, follow these steps:

Step 1: Choose the King

There are 4 kings in a deck of 52 cards. We want exactly 1 king in our 5-card combination, so we first choose 1 king from the 4 available kings.

The number of ways to choose 1 king from 4 is calculated using combinations: $\binom{4}{1} = 4$

Step 2: Choose the Remaining 4 Cards

After selecting 1 king, we need to choose 4 more cards from the remaining 48 cards (since there are 52 cards in total, and we have already used 1 king).

The number of ways to choose 4 cards from 48 is: $\binom{48}{4} = \frac{48 \times 47 \times 46 \times 45}{4 \times 3 \times 2 \times 1} = 194,580$

Step 3: Multiply the Two Results

To find the total number of possible 5-card combinations with exactly 1 king, multiply the result of choosing the king by the result of choosing the other 4 cards: $4 \times 194,580 = 778,320$

Final Answer:

The total number of 5-card combinations with exactly 1 king is 778,320.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How many 5-card combinations have exactly 2 kings?
2. How many 5-card combinations contain no kings at all?
3. What is the probability of drawing exactly 1 king in a 5-card hand?
4. How many 5-card combinations have at least 1 king?
5. How many 5-card hands contain exactly 1 ace and 1 king?

Tip: When dealing with card problems, combinations (denoted as $\binom{n}{k}$) are often used because the order of selection does not matter.