The problem states:

• There are 5 freshmen, 2 sophomores, 4 juniors, and 3 seniors forming a committee.
• A group of 4 people is needed to work on the social media account.
• We need to calculate the probability that the group contains at most 2 juniors.

Let's break this problem down:

Step 1: Total number of ways to select 4 people

There are a total of $5 + 2 + 4 + 3 = 14$ people available. The number of ways to choose a group of 4 people from 14 is given by the combination formula $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n$ is the total number of people and $k$ is the number of people to choose.

So, the total number of ways to select 4 people from 14 is: $\binom{14}{4}$

Step 2: Number of ways to select 4 people with at most 2 juniors

We can have:

• 0 juniors in the group
• 1 junior in the group
• 2 juniors in the group

For each of these cases, we will compute the number of ways to choose the juniors and the remaining people from the non-juniors (freshmen, sophomores, and seniors).

• Case 1: 0 juniors
  Choose 0 juniors from 4 juniors and 4 people from the 10 non-juniors: $\binom{4}{0} \times \binom{10}{4}$

• Case 2: 1 junior
  Choose 1 junior from 4 juniors and 3 people from the 10 non-juniors: $\binom{4}{1} \times \binom{10}{3}$

• Case 3: 2 juniors
  Choose 2 juniors from 4 juniors and 2 people from the 10 non-juniors: $\binom{4}{2} \times \binom{10}{2}$

Step 3: Probability

The probability is the ratio of the favorable outcomes (total number of ways to choose a group with at most 2 juniors) to the total number of outcomes (total number of ways to choose 4 people from 14).

Let me calculate this now.The total number of ways to select 4 people from 14 is 1001.

The number of favorable outcomes, where the group has at most 2 juniors, is 960.

Thus, the probability that the group contains at most 2 juniors is approximately 0.959 (rounded to four decimal places).

Would you like further details or have any other questions?

Here are 5 related questions for practice:

1. What is the probability that there are exactly 3 juniors in the group?
2. What is the probability of selecting a group with at least 1 sophomore?
3. How many ways can we form a group with exactly 1 freshman and 1 senior?
4. What is the probability that no freshmen are selected in the group of 4?
5. If the group is expanded to 5 people, how does the probability of having at most 2 juniors change?

Tip: When dealing with combinations, always consider breaking down complex problems into cases to make calculations manageable.