Part 1: Total number of ways to choose 17 students from 35

To choose a dance committee of 17 students from the total pool of 35 students (8 freshmen + 9 sophomores + 10 juniors + 8 seniors), the number of ways can be found using the combination formula:

$\binom{n}{r} = \frac{n!}{r!(n-r)!}$

Where $n = 35$ (total students) and $r = 17$ (committee size). The solution already provides the total as 3,262,623,600, so we will focus on the next part.

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Part 2: Number of ways to choose 5 freshmen, 6 sophomores, 2 juniors, and 4 seniors

Here, we have to choose specific numbers from each class:

• 5 freshmen from 8,
• 6 sophomores from 9,
• 2 juniors from 10, and
• 4 seniors from 8.

We use the combination formula for each group:

$\binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$

$\binom{9}{6} = \frac{9!}{6!(9-6)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$

$\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2} = 45$

$\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70$

Now, multiply the results together to find the total number of ways to form this specific committee:

$56 \times 84 \times 45 \times 70 = 14,832,000$

Final Answer:

The number of ways to choose a dance committee consisting of 5 freshmen, 6 sophomores, 2 juniors, and 4 seniors is $14,832,000$.

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Do you want further details or have any questions?

Here are 5 related questions to consider:

1. How would the number change if the committee required an equal number of students from each class?
2. What if there were no restrictions on the number of freshmen or sophomores in the committee?
3. Can you determine how many ways to form a committee of 10 students with at least one from each class?
4. How do combinations differ from permutations in problems like this?
5. How would the solution change if the number of juniors was greater than 10?

Tip: When solving combination problems, remember that order doesn't matter; this is key in distinguishing between combinations and permutations.