There are n socks in a drawer, of which 3 socks are red. If 2 of the socks are chosen randomy and the probability that both selected socks are red is `(1)/(2)` then what is the value of n ?

Options

A. 3
B. 4
C. 5
D. 6

Answer

To solve the problem, we need to find the value of \( n \) given that the probability of selecting 2 red socks from a total of \( n \) socks (of which 3 are red) is \( \frac{1}{2} \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have \( n \) socks in total, out of which 3 are red. We need to find the total number of socks \( n \) such that the probability of picking 2 red socks is \( \frac{1}{2} \).

2. **Setting Up the Probability**:
The probability of selecting 2 red socks from the 3 red socks can be expressed using combinations:
\[ P(\text{both socks are red}) = \frac{\text{Number of ways to choose 2 red socks}}{\text{Total ways to choose 2 socks from } n} \]
This can be written as:
\[ P(\text{both socks are red}) = \frac{\binom{3}{2}}{\binom{n}{2}} \]

3. **Calculating the Combinations**:
The number of ways to choose 2 red socks from 3 is:
\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3 \]
The total number of ways to choose 2 socks from \( n \) is:
\[ \binom{n}{2} = \frac{n!}{2!(n-2)!} = \frac{n(n-1)}{2} \]

4. **Setting Up the Equation**:
We know from the problem that:
\[ \frac{3}{\frac{n(n-1)}{2}} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 3 \cdot 2 = 1 \cdot \frac{n(n-1)}{2} \]
Simplifying this results in:
\[ 6 = \frac{n(n-1)}{2} \]
Multiplying both sides by 2:
\[ 12 = n(n-1) \]

5. **Rearranging the Equation**:
Rearranging gives us a quadratic equation:
\[ n^2 - n - 12 = 0 \]

6. **Factoring the Quadratic**:
We can factor this equation:
\[ (n - 4)(n + 3) = 0 \]
This gives us two potential solutions:
\[ n - 4 = 0 \quad \text{or} \quad n + 3 = 0 \]
Thus, \( n = 4 \) or \( n = -3 \).

7. **Selecting the Valid Solution**:
Since \( n \) must be a positive integer (as it represents the number of socks), we discard \( n = -3 \).

8. **Conclusion**:
Therefore, the value of \( n \) is:
\[ n = 4 \]

### Final Answer:
The total number of socks \( n \) is \( 4 \).