From a group of 3 freshmen, 4 sophomores, 4 juniors, and 3 seniors a committee of size 4 is randomly selected. Find the probability that ,the committee will consist of i. one from each class 
i. 2 sophomores and 2 juniors               ii. Only sophomore or junior


Let's solve the probability problems one by one.

### Given:
- 3 freshmen
- 4 sophomores
- 4 juniors
- 3 seniors

Total students = \(3 + 4 + 4 + 3 = 14\)

A committee of 4 is to be selected.

### Total Possible Combinations:
The total number of ways to choose 4 students out of 14 is calculated using the combination formula:

\[
\text{Total ways} = \binom{14}{4} = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} = 1001
\]

### Part i: Probability that the committee consists of one from each class

To satisfy this condition, we need to choose:
- 1 freshman out of 3
- 1 sophomore out of 4
- 1 junior out of 4
- 1 senior out of 3

The number of favorable combinations is:

\[
\text{Favorable ways} = \binom{3}{1} \times \binom{4}{1} \times \binom{4}{1} \times \binom{3}{1} = 3 \times 4 \times 4 \times 3 = 144
\]

So, the probability is:

\[
P(\text{one from each class}) = \frac{144}{1001}
\]

### Part ii: Probability that the committee consists of 2 sophomores and 2 juniors

To satisfy this condition, we need to choose:
- 2 sophomores out of 4
- 2 juniors out of 4

The number of favorable combinations is:

\[
\text{Favorable ways} = \binom{4}{2} \times \binom{4}{2} = \frac{4 \times 3}{2 \times 1} \times \frac{4 \times 3}{2 \times 1} = 6 \times 6 = 36
\]

So, the probability is:

\[
P(\text{2 sophomores and 2 juniors}) = \frac{36}{1001}
\]

### Part iii: Probability that the committee consists of only sophomores or juniors

For this condition, we need to choose all 4 members from the group of sophomores and juniors (8 in total).

The number of favorable combinations is:

\[
\text{Favorable ways} = \binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70
\]

So, the probability is:

\[
P(\text{only sophomores or juniors}) = \frac{70}{1001}
\]

### Summary of Probabilities:
1. **One from each class**: \( \frac{144}{1001} \)
2. **2 sophomores and 2 juniors**: \( \frac{36}{1001} \)
3. **Only sophomores or juniors**: \( \frac{70}{1001} \)

Would you like further details on any of these calculations, or have any questions?

Here are 5 related questions you might find interesting:
1. What is the probability of selecting all freshmen in the committee?
2. How would the probabilities change if there were 5 sophomores instead of 4?
3. What is the probability of selecting at least one senior in the committee?
4. How many ways can you form a committee of 4 members such that no freshmen are included?
5. What is the probability that the committee will consist of 3 juniors and 1 senior?

**Tip:** Remember, when calculating probabilities for combinations, breaking down the problem into smaller parts helps in ensuring accuracy.

Explore the probability of forming committees from freshmen, sophomores, juniors, and seniors in a college-level math problem. Calculate probabilities for committees consisting of one member from each class, 2 sophomores and 2 juniors, and only sophomores or juniors.

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