Question

six sophomores and 14 freshmen are competing for two alternate positions on the debate team. which expression represents the probability that both students chosen are sophomores?
$\frac{_{6}c_{2}}{_{20}c_{2}}$
$\frac{_{6}p_{2}}{_{20}p_{2}}$
$\frac{(_{20}c_{6})(_{19}c_{5})}{_{20}c_{2}}$
$\frac{(_{20}p_{6})(_{19}p_{5})}{_{20}p_{2}}$

Explanation:

Step1: Calculate total number of students

Total number of students \(n = 6+14=20\).

Step2: Use combination formula for probability

Probability \(P=\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\).
Number of ways to choose \(2\) sophomores out of \(6\) is \(_{6}C_{2}=\frac{6!}{2!(6 - 2)!}=\frac{6\times5}{2\times 1}\).
Number of ways to choose \(2\) students out of \(20\) is \(_{20}C_{2}=\frac{20!}{2!(20 - 2)!}=\frac{20\times19}{2\times1}\).
Since the order of choosing the two - alternate positions does not matter (as both are alternate positions, not distinct like president - vice - president), we use combinations.

Answer:

\(\frac{_{6}C_{2}}{_{20}C_{2}}\) (First option)