Three students will represent your school at a conference at the state capitol. A drawing will be held to select the three representatives from a group of five finalists. If you and you friend are both finalists, what are the chances that both of your names will be drawn?

There are 5 choose 3 ways to draw 3 members.

There are 3 choose 2 ways to draw the two specified people.
So the probability is
P(2)=(3,2)/(5,3)=3/10

To find the probability of both your names being drawn, we need to determine the total number of possible outcomes and the number of favorable outcomes.

Total number of outcomes: The first representative can be selected from any of the five finalists. The second representative can be selected from the remaining four finalists. The third representative can be selected from the remaining three finalists. Therefore, the total number of outcomes is:

5 * 4 * 3 = 60

Favorable outcomes: Since you and your friend are both finalists, your names need to be selected together along with one more student. The first representative is selected from the two finalists (you or your friend). The second representative is selected from the remaining three finalists. Therefore, the number of favorable outcomes is:

2 * 3 = 6

Probability: The probability of both your names being drawn is the ratio of favorable outcomes to total outcomes:

P(both your names being drawn) = favorable outcomes / total outcomes

P(both your names being drawn) = 6 / 60

P(both your names being drawn) = 1 / 10

Therefore, the chances that both of your names will be drawn are 1 in 10, or 1/10.

To calculate the chances that both you and your friend will be drawn as representatives, you need to determine the probability of each drawing event.

Since there are 5 finalists in total, the probability of drawing your name in the first round is 1 out of 5, or 1/5. After your name has been drawn, there will be 4 finalists remaining, including your friend. Therefore, the probability of drawing your friend's name in the second round is 1 out of 4, or 1/4.

To calculate the probability of both events occurring, you multiply the probabilities together:

(1/5) * (1/4) = 1/20

Therefore, the chances that both your names will be drawn as representatives are 1 out of 20, or 1/20.