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? StartFraction (6 C 1) (5 C 1) Over 20 C 2 EndFraction StartFraction (6 P 1) (5 P 1) Over 20 P 2 EndFraction StartFraction (20 C 6) (19 C 5) Over 20 C 2 EndFraction StartFraction (20 P 6) (19 P 5) Over 20 P 2 EndFraction

Answer 1

(6C1)(5C1)/20C2

Explanation:

Was right on egde

Answer 2

Choose the first alternative

`\displaystyle P=\frac{_(1)^(6)\textrm{C}\ _(1)^(5)\textrm{C}}{_(2)^(20)\textrm{C}}`

Explanation:

Probabilities

The requested probability can be computed as the ratio between the number of ways to choose two sophomores in alternate positions
`(N_s)` and the total number of possible choices
`(N_t)`, i.e.

`\displaystyle P=(N_s)/(N_t)`

There are 6 sophomores and 14 freshmen to choose from each separate set. There are 20 students in total

We'll assume the positions of the selections are NOT significative, i.e. student A/student B is the same as student B/student A.

To choose 2 sophomores out of the 6 available, the first position has 6 elements to choose from, the second has now only 5

`_(1)^(6)\textrm{C}\ _(1)^(5)\textrm{C} \text{ ways to do it}`

The total number of possible choices is

`_(2)^(20)\textrm{C} \text{ ways to do it}`

The probability is then

`\boxed{\displaystyle P=\frac{_(1)^(6)\textrm{C}\ _(1)^(5)\textrm{C}}{_(2)^(20)\textrm{C}}}`

Choose the first alternative