II There are 32 students in a class 8 freshman 8 sophomores

II: There are 32 students in a class: 8 freshman, 8 sophomores, 8 juniors and 8 seniors. We are selecting 5 of them at random. You may leave all answers in combinatoral form.

1. What is the probability that they are all seniors?

2. What is the probability that they are all in the same class?

3. What is the probability of getting at least one freshman?

4. What is the probability that all four class are represented?

Solution

1.

There are 32C5 ways to choose any 5 students, and 8C5 ways to choose 5 seniors.

P(all seniors) = 8C5 / 32C5 [ANSWER]

**********

2.

There are 8C5 ways to choose all freshman.
There are 8C5 ways to choose all sophomore.
There are 8C5 ways to choose all junior.
There are 8C5 ways to choose all senior.

P(all in same class) = [4 * (8C5)] / (32C5) [ANSWER]

************

3.

There are 24C5 ways to choose 5 non-fresmen.

Thus,

P(at least one freshman) = 1 - P(no freshman)

= 1 - (24C5)/(32C5) [ANSWER]

*******************

4.

We first fix 4 students to be one of each class. There are 8^4 ways to do it. Then we can choose the last one in 28 ways.

Thus,

P(at least one of each class) = [8^4 * 28] / (32C5) [ANSWER]