10 freshmen, 9 sophomores, 7 juniors, and 6 seniors are eligible to be on a committee. If a committee of 14 students is chosen at random, what is the probability that it is made up of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors?
Round your answer to 6 decimal places as needed.

To solve this problem, we need to calculate the probability of selecting a committee with a specific composition (2 freshmen, 3 sophomores, 4 juniors, and 5 seniors) from a group of students.

### Step 1: Determine the total number of possible ways to choose 14 students
The total number of eligible students is:
\[
10 \text{ (freshmen)} + 9 \text{ (sophomores)} + 7 \text{ (juniors)} + 6 \text{ (seniors)} = 32 \text{ students}.
\]
The number of ways to choose 14 students from 32 is given by the combination formula:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
So, the total number of ways to select 14 students from 32 is:
\[
\binom{32}{14}
\]

### Step 2: Calculate the number of favorable outcomes
The number of ways to select:
- 2 freshmen from 10 freshmen: \(\binom{10}{2}\)
- 3 sophomores from 9 sophomores: \(\binom{9}{3}\)
- 4 juniors from 7 juniors: \(\binom{7}{4}\)
- 5 seniors from 6 seniors: \(\binom{6}{5}\)

Thus, the total number of favorable outcomes is:
\[
\binom{10}{2} \times \binom{9}{3} \times \binom{7}{4} \times \binom{6}{5}
\]

### Step 3: Calculate the probability
The probability is the ratio of the number of favorable outcomes to the total number of outcomes:
\[
\text{Probability} = \frac{\binom{10}{2} \times \binom{9}{3} \times \binom{7}{4} \times \binom{6}{5}}{\binom{32}{14}}
\]
Now, let's compute these values.The probability that a randomly chosen committee of 14 students will consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors is approximately \( 0.001684 \) (rounded to six decimal places).

Do you want more details or have any questions?

Here are 5 related questions:
1. How would the probability change if the total committee size were 15 instead of 14?
2. What is the probability of selecting exactly 4 freshmen and 5 seniors?
3. How does changing the composition (e.g., choosing more sophomores) affect the probability?
4. What is the total number of possible committees of 14 students from 32 students?
5. How do you calculate combinations using factorials?

**Tip**: When calculating probabilities in combinatorics, always identify both the total possible outcomes and the number of favorable outcomes to build the correct probability ratio.

10 freshmen, 9 sophomores, 7 juniors, and 6 seniors are eligible to be on a committee. If a committee of 14 students is chosen at random, what is the probability that it is made up of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors? Round your answer to 6 decimal places as needed.

Calculate the probability of randomly selecting a committee of 14 students consisting of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors from a total of 32 students. This problem involves combinatorics and probability theory and is suitable for Grades 10-12.

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