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

There are 6 sophomores and 14 freshmen, so the total number of students is $6 + 14=20$.

Step2: Determine the total number of ways to choose 2 students

The number of ways to choose 2 students out of 20 for the two - alternate positions 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 ${20}C{2}$.

Step3: Determine the number of ways to choose 2 sophomores

The number of ways to choose 2 sophomores out of 6 is given by the combination formula with $n = 6$ and $r = 2$, which is ${6}C{2}$.

Step4: Calculate the probability

The probability 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 the probability is $\frac{{6}C{2}}{{20}C{2}}$.

Answer:

$\frac{{6}C{2}}{{20}C{2}}$