To answer the question, we need to determine the number of black pens in the bag.
Given:
- Number of red pens = 3
- Number of blue pens = 7
- Probability of picking a red pen = [tex]\(\frac{1}{6}\)[/tex]
- Let the number of black pens be [tex]\(x\)[/tex]
The total number of pens in the bag is the sum of red, blue, and black pens. Therefore, the total number of pens is:
[tex]\[ \text{Total number of pens} = 3 + 7 + x = 10 + x \][/tex]
The probability of picking a red pen is given by the ratio of the number of red pens to the total number of pens. Therefore, we have the following equation for the probability:
[tex]\[ \frac{3}{10 + x} = \frac{1}{6} \][/tex]
Next, we solve this equation to find the value of [tex]\(x\)[/tex]:
1. Cross-multiply to solve for [tex]\(x\)[/tex]:
[tex]\[ 3 \cdot 6 = (10 + x) \cdot 1 \][/tex]
[tex]\[ 18 = 10 + x \][/tex]
2. Isolate [tex]\(x\)[/tex] by subtracting 10 from both sides of the equation:
[tex]\[ 18 - 10 = x \][/tex]
[tex]\[ x = 8 \][/tex]
So, the number of black pens is [tex]\(8\)[/tex]. Thus, the correct answer is:
[tex]\[ \boxed{8} \][/tex]
