find the number of 5 card combinations out of a deck of 52 cards if each selection of 5 cards has exactly 1 king

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.

Find the number of 5 card combinations out of a deck of 52 cards if each selection of 5 cards has exactly 1 king.

Learn how to calculate the number of 5-card combinations from a deck of 52 cards where exactly 1 card is a king. This solution involves combinatorics, applying the combination formula C(n, k) to find the number of ways to choose 1 king and 4 non-king cards. This problem is suitable for students in Grades 9-12.

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