A school dance committee is made up of 3 freshman, 2 sophomores, 5 juniors, and 3 seniors.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?

To solve this problem, we need to determine how many ways we can arrange the members of the committee in a row, given that the freshman, sophomores, and juniors must sit together by their respective grades, while the seniors can sit wherever they like.

### Step 1: Break down the number of people by grade
- Freshmen: 3
- Sophomores: 2
- Juniors: 5
- Seniors: 3

### Step 2: Treat each grade group as a block
Since the freshmen, sophomores, and juniors must sit together by grade, we treat each group as a "block." The total number of blocks is:
- Freshman block
- Sophomore block
- Junior block
- Seniors (they are not constrained)

We will first arrange the three blocks (Freshmen, Sophomores, Juniors) in a fixed way because they must sit together. Therefore, there are \( 3! \) ways to arrange these blocks relative to each other.

### Step 3: Arrange individuals within each block
For each block, the individuals can be arranged in any order:
- Freshmen: There are 3 freshmen, and they can be arranged among themselves in \( 3! \) ways.
- Sophomores: There are 2 sophom

A school dance committee is made up of 3 freshman, 2 sophomores, 5 juniors, and 3 seniors. 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?

Learn how to solve a combinatorics problem involving a school dance committee with 3 freshmen, 2 sophomores, 5 juniors, and 3 seniors. The freshmen, sophomores, and juniors must sit together by grade, while the seniors can sit anywhere. This problem covers permutation concepts and factorial calculations. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation