How many ways can you select 4 cards from a deck of 52 cards if the selection must include exactly 2 spades?

To solve this problem, we need to select 4 cards from a deck of 52 cards with the condition that exactly 2 of those cards are spades.

### Step 1: Understanding the Deck and Requirements
- A standard deck has 52 cards, which includes 13 cards of each suit (hearts, diamonds, clubs, spades).
- We need to select 4 cards, with exactly 2 being spades.

### Step 2: Choosing 2 Spades
- The number of ways to choose 2 spades from the 13 spades is given by the combination formula:

\[
\text{Number of ways to choose 2 spades} = \binom{13}{2}
\]

Calculating this:

\[
\binom{13}{2} = \frac{13 \times 12}{2 \times 1} = 78
\]

### Step 3: Choosing the Remaining 2 Cards from Non-Spades
- After selecting 2 spades, we need to choose the remaining 2 cards from the non-spade cards. There are 52 cards in total, and 13 of them are spades, so there are \( 52 - 13 = 39 \) non-spade cards.

- The number of ways to choose 2 cards from these 39 non-spade cards is given by:

\[
\text{Number of ways to choose 2 non-spade cards} = \binom{39}{2}
\]

Calculating this:

\[
\binom{39}{2} = \frac{39 \times 38}{2 \times 1} = 741
\]

### Step 4: Calculating the Total Number of Ways
- To find the total number of ways to select 4 cards with exactly 2 spades, we multiply the number of ways to choose the spades by the number of ways to choose the non-spades:

\[
\text{Total number of ways} = \binom{13}{2} \times \binom{39}{2}
\]

\[
\text{Total number of ways} = 78 \times 741 = 57,798
\]

### Final Answer
**There are 57,798 ways to select 4 cards from a deck of 52 cards if the selection must include exactly 2 spades.**

Would you like more details or have any questions?

### Related Questions:
1. How many ways are there to select 4 cards if the selection must include exactly 3 spades?
2. How many ways are there to select 4 cards if all 4 must be spades?
3. What is the probability of drawing exactly 2 spades when 4 cards are drawn from a deck?
4. How many ways can you select 5 cards from a deck of 52 if the selection must include at least 2 hearts?
5. How many ways are there to select 4 cards from a deck of 52 if no card can be a club?

### Tip:
When dealing with problems involving specific conditions on selection (like certain numbers of spades), breaking the problem into smaller, manageable parts and using combinations can help simplify the solution process.

Explore the calculation of selecting 4 cards from a deck of 52 cards with exactly 2 spades using combinatorial mathematics. Understand the combination formula, and learn how to apply it step-by-step. Suitable for advanced high school students.

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