Let's break down the problem step by step.
### Step 1: Determine Total Number of Marbles
First, we need to calculate the total number of marbles in the bag:
- Red marbles: 10
- Yellow marbles: 15
- Green marbles: 5
- Blue marbles: 20
Total marbles = [tex]\(10 + 15 + 5 + 20 = 50 \)[/tex]
### Step 2: Choose 1 Red Marble and 1 Blue Marble
Next, we need to determine the number of ways to choose 1 red marble and 1 blue marble:
- The number of ways to choose 1 red marble from 10 red marbles = [tex]\(\binom{10}{1}\)[/tex]
- The number of ways to choose 1 blue marble from 20 blue marbles = [tex]\(\binom{20}{1}\)[/tex]
The total number of ways to choose 1 red marble and 1 blue marble is:
[tex]\[ \binom{10}{1} \times \binom{20}{1} = 10 \times 20 = 200 \][/tex]
### Step 3: Total Number of Ways to Choose Any 2 Marbles
We need to calculate the total number of ways to choose any 2 marbles out of the 50 marbles. This can be calculated using the combination formula:
[tex]\[ \binom{50}{2} = \frac{50!}{2!(50-2)!} = \frac{50 \times 49}{2 \times 1} = 1225 \][/tex]
### Step 4: Computing the Probability
The probability of drawing 1 red marble and 1 blue marble is the ratio of the number of favorable outcomes to the total number of outcomes:
[tex]\[ \text{Probability} = \frac{\text{Number of ways to choose 1 red and 1 blue}}{\text{Total number of ways to choose any 2 marbles}} = \frac{200}{1225} \][/tex]
When calculated, this fraction simplifies, but let's focus on the expression forms given in the options.
### Step 5: Matching the Expression
From the given options, the expression that represents the probability as calculated is:
[tex]\[ \frac{\left(\binom{10}{1}\right)\left(\binom{20}{1}\right)}{\binom{50}{2}} \][/tex]
This matches with:
[tex]\[ \frac{\left({ }_{10} C_1\right)\left({ }_{20} C_1\right)}{{ }_{50} C_2} \][/tex]
Thus, the correct expression is:
[tex]\[ \frac{\left({ }_{10} C_1\right)\left({ }_{20} C_1\right)}{{ }_{50} C_2} \][/tex]
