There are two red, two blue, two white, and certain number (greater than 0) of green socks n a drawer. If two socks are taken at random from the drawer4 without replacement, the probability that they are of the same color is 1/5, then the number of green socks are ________.

Explanation

To solve the problem step by step, we will follow the outlined reasoning from the video transcript.

### Step 1: Define Variables
Let \( x \) be the number of green socks in the drawer. We know that there are:
- 2 red socks
- 2 blue socks
- 2 white socks
- \( x \) green socks

### Step 2: Calculate Total Number of Socks
The total number of socks in the drawer is:
\[ \text{Total socks} = 2 \text{ (red)} + 2 \text{ (blue)} + 2 \text{ (white)} + x \text{ (green)} = 6 + x \]

### Step 3: Calculate Total Outcomes
When we draw 2 socks from \( 6 + x \) socks, the total number of ways to choose 2 socks is given by the combination formula:
\[ \text{Total outcomes} = \binom{6+x}{2} = \frac{(6+x)(5+x)}{2} \]

### Step 4: Calculate Favorable Outcomes
The favorable outcomes for drawing 2 socks of the same color can be calculated as follows:
1. For red socks: \( \binom{2}{2} = 1 \)
2. For blue socks: \( \binom{2}{2} = 1 \)
3. For white socks: \( \binom{2}{2} = 1 \)
4. For green socks: \( \binom{x}{2} = \frac{x(x-1)}{2} \)

Thus, the total favorable outcomes are:
\[ \text{Favorable outcomes} = 1 + 1 + 1 + \frac{x(x-1)}{2} = 3 + \frac{x(x-1)}{2} \]

### Step 5: Set Up the Probability Equation
According to the problem, the probability of drawing two socks of the same color is given as \( \frac{1}{5} \). Therefore, we can set up the equation:
\[ \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{1}{5} \]
This gives us:
\[ \frac{3 + \frac{x(x-1)}{2}}{\frac{(6+x)(5+x)}{2}} = \frac{1}{5} \]

### Step 6: Cross Multiply to Solve for \( x \)
Cross multiplying gives us:
\[ 5 \left(3 + \frac{x(x-1)}{2}\right) = (6+x)(5+x) \]
Expanding both sides:
\[ 15 + \frac{5x(x-1)}{2} = 30 + 11x + x^2 \]

### Step 7: Eliminate the Fraction
To eliminate the fraction, multiply the entire equation by 2:
\[ 30 + 5x(x-1) = 60 + 22x + 2x^2 \]
This simplifies to:
\[ 30 + 5x^2 - 5x = 60 + 22x + 2x^2 \]

### Step 8: Rearrange the Equation
Rearranging gives:
\[ 5x^2 - 2x^2 - 5x - 22x + 30 - 60 = 0 \]
This simplifies to:
\[ 3x^2 - 27x - 30 = 0 \]

### Step 9: Factor or Use Quadratic Formula
We can simplify this equation:
\[ x^2 - 9x - 10 = 0 \]
Factoring gives:
\[ (x - 10)(x + 1) = 0 \]
Thus, \( x = 10 \) or \( x = -1 \). Since \( x \) must be greater than 0, we have:
\[ x = 10 \]

### Conclusion
The number of green socks is \( \boxed{10} \).