To determine the probability that one of the drawn marbles is red and the other is blue, let's follow these steps:
1. Step 1: Calculate the total number of marbles.
[tex]\[
\text{Total number of marbles} = 10 \text{ (red)} + 15 \text{ (yellow)} + 5 \text{ (green)} + 20 \text{ (blue)} = 50
\][/tex]
2. Step 2: Calculate the total number of ways to draw 2 marbles from the 50 marbles.
This is done using the combination formula [tex]\( _{n} C _{k} \)[/tex] which represents the number of ways to choose [tex]\( k \)[/tex] items from [tex]\( n \)[/tex] items without regard to order:
[tex]\[
_{50} C _{2} = \frac{50!}{2!(50-2)!}
\][/tex]
3. Step 3: Determine the number of favorable combinations where one marble is red and the other is blue.
For one red marble, we have:
[tex]\[
_{10} C _{1} = 10
\][/tex]
For one blue marble, we have:
[tex]\[
_{20} C _{1} = 20
\][/tex]
4. Step 4: Combine these two events (drawing one red and one blue marble). The events are independent, so we multiply the combinations:
[tex]\[
\text{Number of favorable outcomes} = 10 \times 20 = 200
\][/tex]
5. Step 5: Put all together in a probability expression:
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{_{10} C _{1} \times _{20} C _{1}}{{ }_{50} C _{2}}
\][/tex]
So, the expression that represents the probability that one of the marbles is red and the other is blue is:
[tex]\[
\frac{\left({ }_{10} C _1\right)\left({ }_{20} C _1\right)}{{ }_{50} C _2}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{\left({ }_{10} C _1\right)\left({ }_{20} C _1\right)}{{ }_{50} C _2}}
\][/tex]
