1) Total ways to choose 4 cards from a deck of 52:

The total number of ways to choose 4 cards from a deck of 52 cards is given by the combination formula:

$\binom{n}{r} = \frac{n!}{r!(n-r)!}$

For $n = 52$ and $r = 4$:

$\binom{52}{4} = \frac{52!}{4!(52-4)!} = \frac{52 \times 51 \times 50 \times 49}{4 \times 3 \times 2 \times 1} = 270725$

Thus, there are 270,725 ways to choose 4 cards from a deck of 52.

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2) Ways to choose 4 cards with 2 red and 2 black:

• There are 26 red cards and 26 black cards in a standard deck.
• To have exactly 2 red cards and 2 black cards, we calculate the combinations separately for red and black cards:

Choosing 2 red cards out of 26: $\binom{26}{2} = \frac{26 \times 25}{2 \times 1} = 325$

Choosing 2 black cards out of 26: $\binom{26}{2} = \frac{26 \times 25}{2 \times 1} = 325$

Total ways to choose 2 red and 2 black cards: $\binom{26}{2} \times \binom{26}{2} = 325 \times 325 = 105625$

Thus, there are 105,625 ways to choose 4 cards with exactly 2 red and 2 black.

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Summary:

1. Total ways to draw 4 cards: 270,725
2. Total ways to draw 4 cards with 2 red and 2 black: 105,625

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Would you like a detailed step-by-step explanation of these calculations or any further clarifications?

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Relative Questions:

1. How can we calculate combinations when conditions are applied, such as specific suits or numbers?
2. What is the probability of drawing 2 red and 2 black cards in a 4-card draw?
3. Can we calculate similar probabilities for hands in poker?
4. How do these concepts apply to drawing cards with replacement?
5. What are the key differences between permutations and combinations?

Tip:

Always distinguish between order matters (permutations) and order does not matter (combinations) when solving probability or counting problems.