From 4 red balls, 2 white balls and 4 black balls, four balls are selected. The probability of getting 2 red balls is p, then the value of 7p

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we need to calculate the probability \( p \) of selecting 2 red balls when we randomly select 4 balls from a total of 10 balls (4 red, 2 white, and 4 black).

### Step-by-Step Solution:

1. **Identify Total Balls**:
We have:
- 4 Red balls
- 2 White balls
- 4 Black balls
- **Total = 4 + 2 + 4 = 10 balls**

2. **Determine the Required Selection**:
We want to find the probability of selecting exactly 2 red balls when selecting 4 balls. This means we need to consider the other 2 balls which can be either white or black.

3. **Cases for the Remaining Balls**:
The remaining 2 balls can be:
- 2 White balls
- 2 Black balls
- 1 White and 1 Black ball

4. **Calculate Probabilities for Each Case**:
We will calculate the probability for each case and then sum them up.

- **Case 1**: 2 Red and 2 White
\[ \text{Number of ways to choose 2 red from 4} = \binom{4}{2} = 6 \]
\[ \text{Number of ways to choose 2 white from 2} = \binom{2}{2} = 1 \]
\[ \text{Total for this case} = 6 \times 1 = 6 \]

- **Case 2**: 2 Red and 2 Black
\[ \text{Number of ways to choose 2 red from 4} = \binom{4}{2} = 6 \]
\[ \text{Number of ways to choose 2 black from 4} = \binom{4}{2} = 6 \]
\[ \text{Total for this case} = 6 \times 6 = 36 \]

- **Case 3**: 2 Red, 1 White, and 1 Black
\[ \text{Number of ways to choose 2 red from 4} = \binom{4}{2} = 6 \]
\[ \text{Number of ways to choose 1 white from 2} = \binom{2}{1} = 2 \]
\[ \text{Number of ways to choose 1 black from 4} = \binom{4}{1} = 4 \]
\[ \text{Total for this case} = 6 \times 2 \times 4 = 48 \]

5. **Total Favorable Outcomes**:
Now, we sum the outcomes from all cases:
\[ \text{Total favorable outcomes} = 6 + 36 + 48 = 90 \]

6. **Total Possible Outcomes**:
The total ways to choose any 4 balls from 10:
\[ \text{Total outcomes} = \binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210 \]

7. **Calculate Probability \( p \)**:
\[ p = \frac{\text{Total favorable outcomes}}{\text{Total outcomes}} = \frac{90}{210} = \frac{3}{7} \]

8. **Calculate \( 7p \)**:
\[ 7p = 7 \times \frac{3}{7} = 3 \]

### Final Answer:
The value of \( 7p \) is \( \boxed{3} \).