**A school dance committee is made up of 3 freshman, 3 sophomores, 4 juniors, and 3 seniors.**

**How many ways are there to sit the committee in a row at a meeting if the students must sit together by grade?**
Correct

**How many ways are there to sit the committee in a row at a meeting if the freshman,sophomores, and juniors must sit by grade, but the seniors can sit wherever they want?**

### 1. Committee Sits Together by Grade

We are given four groups: 3 freshmen, 3 sophomores, 4 juniors, and 3 seniors.

#### Step 1: Arranging Students Within Each Grade
- Freshmen: \( 3! \) ways to arrange.
- Sophomores: \( 3! \) ways to arrange.
- Juniors: \( 4! \) ways to arrange.
- Seniors: \( 3! \) ways to arrange.

Thus, the number of ways to arrange students within their grades is:
\[
3! \times 3! \times 4! \times 3! = 6 \times 6 \times 24 \times 6 = 5184
\]

#### Step 2: Arranging the Grade Groups Themselves
There are 4 grade groups, and they must sit together as blocks. We can arrange these 4 groups in \( 4! \) ways:
\[
4! = 24
\]

#### Total Number of Arrangements:
Thus, the total number of ways to arrange the committee when students must sit together by grade is:
\[
4! \times 3! \times 3! \times 4! \times 3! = 24 \times 5184 = 124,416
\]

### 2. Freshmen, Sophomores, and Juniors Sit by Grade, but Seniors Can Sit Anywhere

Now, the freshmen, sophomores, and juniors must sit by grade, but seniors can sit anywhere.

#### Step 1: Arranging Freshmen, Sophomores, and Juniors
- The three grades (freshmen, sophomores, juniors) must still sit together as blocks.
- We can arrange these 3 grade blocks in \( 3! \) ways:
  \[
  3! = 6
  \]
- Within each grade, we can arrange the students as follows:
  \[
  3! \times 3! \times 4! = 6 \times 6 \times 24 = 864
  \]
So, the number of ways to arrange the freshmen, sophomores, and juniors is:
\[
3! \times 3! \times 3! \times 4! = 6 \times 864 = 5184
\]

#### Step 2: Placing Seniors
There are 13 total students. Since seniors can sit anywhere, we choose 3 spots for them from 13 available positions. The number of ways to select 3 positions from 13 is:
\[
\binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286
\]
After selecting the positions, we can arrange the 3 seniors in \( 3! = 6 \) ways. Thus, the total number of ways to arrange the seniors is:
\[
\binom{13}{3} \times 3! = 286 \times 6 = 1716
\]

#### Total Number of Arrangements:
Thus, the total number of ways to sit the committee when freshmen, sophomores, and juniors must sit by grade but seniors can sit anywhere is:
\[
5184 \times 1716 = 8,892,864
\]

### Summary:
- When all students must sit by grade: \( 124,416 \) ways.
- When freshmen, sophomores, and juniors must sit by grade, but seniors can sit anywhere: \( 8,892,864 \) ways.

Would you like more details or have any questions?

### Related Questions:
1. What is the number of ways to arrange just the juniors and seniors if they must sit together?
2. How many ways are there to sit the committee if freshmen and sophomores must sit together, but juniors and seniors can sit anywhere?
3. What if only the juniors must sit together, and everyone else can sit anywhere?
4. How many ways can we arrange the committee if no one needs to sit together by grade?
5. What if the seniors must sit at the ends of the row while everyone else can sit anywhere?

### Tip:
When solving problems that involve both grouping and arranging, it's helpful to treat each group as a "block" and count the number of ways to arrange both the blocks and the elements within them.

A school dance committee is made up of 3 freshman, 3 sophomores, 4 juniors, and 3 seniors. How many ways are there to sit the committee in a row at a meeting if the students must sit together by grade? How many ways are there to sit the committee in a row at a meeting if the freshman, sophomores, and juniors must sit by grade, but the seniors can sit wherever they want?

Discover how to calculate the number of ways to arrange a school dance committee made up of freshmen, sophomores, juniors, and seniors. Learn about permutations, combinations, and factorials through detailed solutions to two seating arrangement problems. Suitable for students in Grades 9-12.

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