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

Options

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

Explanation

To solve the problem step by step, we will use the concept of combinations and probability.

### Step 1: Understand the Problem
We have `n` socks in a drawer, out of which 3 are red. We need to find the value of `n` given that the probability of selecting 2 red socks randomly is \( \frac{1}{2} \).

### Step 2: Write the Probability Formula
The probability of selecting 2 red socks from the 3 available red socks can be expressed using combinations:
\[ P(\text{both socks are red}) = \frac{\text{Number of ways to choose 2 red socks}}{\text{Total ways to choose 2 socks from n}} \]

### Step 3: Calculate the Combinations
The number of ways to choose 2 red socks from 3 is given by:
\[ \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2 \times 1}{2 \times 1 \times 1} = 3 \]

The total number of ways to choose 2 socks from `n` socks is:
\[ \binom{n}{2} = \frac{n!}{2!(n-2)!} = \frac{n(n-1)}{2} \]

### Step 4: Set Up the Equation
According to the problem, the probability is given as \( \frac{1}{2} \):
\[ \frac{\binom{3}{2}}{\binom{n}{2}} = \frac{1}{2} \]
Substituting the values we calculated:
\[ \frac{3}{\frac{n(n-1)}{2}} = \frac{1}{2} \]

### Step 5: Cross Multiply to Solve for n
Cross-multiplying gives:
\[ 3 \times 2 = 1 \times n(n-1) \]
This simplifies to:
\[ 6 = n(n-1) \]

### Step 6: Rearrange into a Quadratic Equation
Rearranging gives us:
\[ n^2 - n - 6 = 0 \]

### Step 7: Factor the Quadratic Equation
To factor the quadratic equation:
\[ n^2 - n - 6 = (n - 3)(n + 2) = 0 \]
Setting each factor to zero gives:
1. \( n - 3 = 0 \) → \( n = 3 \)
2. \( n + 2 = 0 \) → \( n = -2 \) (not valid since n must be positive)

### Step 8: Conclusion
Thus, the only valid solution is:
\[ n = 3 \]

### Final Answer
The value of \( n \) is \( 6 \).