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? $\frac{_6c_2}{_{20}c_2}$ $\frac{_6p_2}{_{20}p_2}$ $\frac{(_20c_6)(_{19}c_5)}{_{20}c_2}$ $\frac{(_20p_6)(_{19}p_5)}{_{20}p_2}$

Explanation:

Step1: Understand Combinations vs Permutations

In probability for choosing without order (since the two alternate positions don't have a specific order of selection), we use combinations (\(_nC_r\)) not permutations (\(_nP_r\)). So we can eliminate options with permutations (\(_nP_r\)).

Step2: Total Number of Ways to Choose 2 Students

Total number of students = 6 sophomores + 14 freshmen = 20 students. The number of ways to choose 2 students from 20 is \({}_{20}C_2\).

Step3: Number of Favorable Ways (Choosing 2 Sophomores)

Number of sophomores is 6. The number of ways to choose 2 sophomores from 6 is \({}_6C_2\).

Step4: Probability Formula

Probability is (Number of favorable outcomes) / (Total number of outcomes). So the probability that both are sophomores is \(\frac{{}_6C_2}{{}_{20}C_2}\).

Answer:

\(\frac{{}_6C_2}{{}_{20}C_2}\) (the first option)