six sophomores and 14 freshmen are competing for two alternate positions on the debate team. which…

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?\n\\(\\frac{_{6}c_{2}}{_{20}c_{2}}\\)\n\\(\\frac{_{6}p_{2}}{_{20}p_{2}}\\)\n\\(\\frac{(_{20}c_{6})(_{19}c_{5})}{_{20}c_{2}}\\)\n\\(\\frac{(_{20}p_{6})(_{19}p_{5})}{_{20}p_{2}}\\)

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?\n\\(\\frac{_{6}c_{2}}{_{20}c_{2}}\\)\n\\(\\frac{_{6}p_{2}}{_{20}p_{2}}\\)\n\\(\\frac{(_{20}c_{6})(_{19}c_{5})}{_{20}c_{2}}\\)\n\\(\\frac{(_{20}p_{6})(_{19}p_{5})}{_{20}p_{2}}\\)

Answer

Explanation:

Step1: Calculate total number of students

The total number of students is (6 + 14=20). We want to choose 2 students out of these 20 for the alternate positions. The number of ways to choose 2 students out of 20 is given by the combination formula ({n}C{r}=\frac{n!}{r!(n - r)!}), where (n = 20) and (r=2), so the total number of ways is ({20}C{2}).

Step2: Calculate number of ways to choose 2 sophomores

We have 6 sophomores and we want to choose 2 of them. Using the combination formula with (n = 6) and (r = 2), the number of ways to choose 2 sophomores out of 6 is ({6}C{2}).

Step3: Calculate the probability

The probability (P) that both students chosen are sophomores is the number of favorable outcomes (choosing 2 sophomores) divided by the number of total outcomes (choosing 2 students out of 20). So (P=\frac{{6}C{2}}{{20}C{2}}).

Answer:

(\frac{{6}C{2}}{{20}C{2}}) (the first - option)