There are 5 white and 3 black balls in the box, and 4 are risked. Find the probability that there are 2 white and 2 black balls among the received balls.

Choose one or more answers:

a. 3/7

b. 5/7

c. 4/7

d. 2/7

To find the probability that there are 2 white and 2 black balls among the 4 balls drawn from the box containing 5 white and 3 black balls, we can use combinatorics.

### Step-by-Step Solution:

1. **Total number of ways to choose 4 balls out of 8** (without regard to color):
\[ \binom{8}{4} = \frac{8!}{4!(8-4)!} = 70 \]

2. **Number of ways to choose 2 white balls out of 5**:
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = 10 \]

3. **Number of ways to choose 2 black balls out of 3**:
\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = 3 \]

4. **Number of favorable outcomes (2 white and 2 black)**:
\[ \binom{5}{2} \times \binom{3}{2} = 10 \times 3 = 30 \]

5. **Probability** (favorable outcomes over total outcomes):
\[ P(\text{2 white, 2 black}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{30}{70} = \frac{3}{7} \]

So the probability that there are 2 white and 2 black balls among the received balls is \( \frac{3}{7} \).

### Answer:
The correct choice is:
- a. \(\frac{3}{7}\)

Do you have any questions or would you like further details?

Here are some questions you might want to ask:
1. How do you calculate combinations (n choose k)?
2. What is the difference between combinations and permutations?
3. How would the probability change if there were more balls of each color?
4. Can you explain the concept of independent and dependent events in probability?
5. How do you use the binomial theorem in probability problems?

**Tip:** Always break down probability problems step by step to ensure accuracy in counting the favorable and total outcomes.

Learn how to solve a probability problem involving drawing 4 balls from a box containing 5 white and 3 black balls. Understand the application of combinatorics and the binomial coefficient formula to determine the probability of drawing 2 white and 2 black balls. Suitable for high school students studying probability and combinatorics.

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