Problem 1: Students sit together by grade

We need to figure out how many ways the committee can be seated in a row, given that the students of each grade must sit together as a group.

• Freshman: 3 students can be seated in $3!$ ways.
• Sophomores: 2 students can be seated in $2!$ ways.
• Juniors: 5 students can be seated in $5!$ ways.
• Seniors: 3 students can be seated in $3!$ ways.

Now, treat the 4 grades (Freshman, Sophomores, Juniors, and Seniors) as "blocks." These blocks can be arranged in $4!$ ways.

So, the total number of ways to seat the students is:

$4! \times 3! \times 2! \times 5! \times 3! = 24 \times 6 \times 2 \times 120 \times 6 = 207360$

Problem 2: Freshman, Sophomores, and Juniors must sit by grade, but Seniors can sit anywhere

In this case, the Freshman, Sophomores, and Juniors still need to be seated together by grade. So we handle those three groups first.

1. Freshman: 3 students can be seated in $3!$ ways.
2. Sophomores: 2 students can be seated in $2!$ ways.
3. Juniors: 5 students can be seated in $5!$ ways.

These three grades form a "block," which can be arranged in $3!$ ways.

Next, we consider the Seniors. Since they can sit anywhere, they have 4 open spots (because there are 3 groups and 3 individual seniors). The total number of places to arrange the seniors is the number of permutations of the total number of people.

So the total number of ways to seat everyone is:

$3! \times 3! \times 3! \times 2! \times 5! = 6 \times 6 \times 2 \times 120 = 51840$

Thus, there are 51,840 ways to arrange the committee in this scenario.

---

Would you like further details or have any questions?

Here are 5 questions to explore further:

1. How many ways are there if only the juniors need to sit together by grade?
2. What happens if the seniors can only sit at the two ends?
3. How many ways are there if no restrictions are placed on seating?
4. How many ways are there to choose 2 committee members from the juniors and seniors?
5. How many ways are there to select a president from the 13 members?

Tip: Factorials grow rapidly, so consider the factorial function when counting arrangements!