To find the probability that both students chosen are sophomores, we need to consider the number of ways to choose 2 sophomores out of a total of 6, and the number of ways to choose any 2 students out of a total of 20 students (6 sophomores and 14 freshmen).
Firstly, we need to calculate the number of ways to choose 2 sophomores from the 6 sophomores. This is represented by the combination formula [tex]\(\binom{6}{2}\)[/tex], which indicates the number of ways to choose 2 items from 6 without regard to order. The solution to this combination formula is:
[tex]\[
\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15
\][/tex]
Next, we calculate the total number of ways to choose 2 students from all 20 students. This is represented by the combination formula [tex]\(\binom{20}{2}\)[/tex]:
[tex]\[
\binom{20}{2} = \frac{20!}{2!(20-2)!} = \frac{20 \times 19}{2 \times 1} = 190
\][/tex]
To find the probability that both students chosen are sophomores, we divide the number of ways to choose 2 sophomores by the number of ways to choose any 2 students from the 20. Therefore, the probability is:
[tex]\[
\text{Probability} = \frac{\binom{6}{2}}{\binom{20}{2}} = \frac{15}{190} = \frac{3}{38} \approx 0.07894736842105263
\][/tex]
Hence, the expression that represents the probability that both students chosen are sophomores is:
[tex]\[
\frac{\binom{6}{2}}{\binom{20}{2}}
\][/tex]
So the corresponding answer is:
[tex]\[
\boxed{\frac{{ }_6 C_2}{{ }_{20} C_2}}
\][/tex]
