The problem asks for the probability that a group of 8 people, selected from a committee consisting of 4 freshmen, 4 sophomores, 7 juniors, and 10 seniors, includes exactly 2 people from each class (freshmen, sophomores, juniors, and seniors).

Let's break it down step by step:

Step 1: Determine the total number of ways to choose 8 people from the entire group.

The total number of people in the committee is: $4 \text{ (freshmen)} + 4 \text{ (sophomores)} + 7 \text{ (juniors)} + 10 \text{ (seniors)} = 25 \text{ people}$

The total number of ways to choose 8 people out of 25 is given by the combination formula: $\text{Total combinations} = \binom{25}{8}$

Step 2: Calculate the number of favorable outcomes.

The favorable outcomes are those in which the group has exactly 2 freshmen, 2 sophomores, 2 juniors, and 2 seniors. The number of ways to achieve this is: $\text{Favorable combinations} = \binom{4}{2} \times \binom{4}{2} \times \binom{7}{2} \times \binom{10}{2}$

Step 3: Compute the probability.

The probability is the ratio of the number of favorable outcomes to the total number of outcomes: $\text{Probability} = \frac{\text{Favorable combinations}}{\text{Total combinations}} = \frac{\binom{4}{2} \times \binom{4}{2} \times \binom{7}{2} \times \binom{10}{2}}{\binom{25}{8}}$

Now, let's compute these values.

1. $\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6$
2. $\binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21$
3. $\binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45$
4. $\binom{25}{8}$ is a larger number, so let me compute it.

I'll calculate this probability for you.The probability that the group of 8 people includes exactly 2 freshmen, 2 sophomores, 2 juniors, and 2 seniors is approximately 0.0315.

Would you like more details or have any further questions?

Here are some related questions you might consider:

1. How do you calculate combinations for different group sizes?
2. What is the probability of selecting a group with only one freshman?
3. How does the probability change if there are more seniors in the committee?
4. What is the significance of rounding in probability calculations?
5. How would the probability differ if the total number of people in the committee changed?

Tip: Understanding the combination formula is crucial when dealing with problems involving selections without replacement.