A school dance committee is made up of 3 freshman, 5 sophomores, 2 juniors, and 4 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?
To find the number of ways to arrange the committee in a row at a meeting if the students must sit together by grade, we can treat each grade as a distinct group and calculate the number of ways to arrange these groups.
1. First, let's consider the number of ways to arrange the freshman. Since there are 3 freshmen, there are 3! (3 factorial) ways to arrange them within their group.
2. Similarly, for the sophomores, there are 5! ways to arrange them within their group.
3. For the juniors, there are 2! ways to arrange them within their group.
4. Now, since the different grade groups are arranged consecutively, we can consider the arrangement of the groups as a single set. In other words, we need to arrange the 3 groups: freshman, sophomores, and juniors. There are 3! ways to arrange these groups.
Finally, to find the total number of arrangements, we multiply the number of ways to arrange each group:
Total number of arrangements = 3! * 5! * 2! * 3! = 6 * 120 * 2 * 6 = 86,400
Therefore, there are 86,400 ways to arrange the committee in a row at a meeting if the students must sit together by grade.
For the second question, where the seniors can sit wherever they want, we only need to consider the arrangement of the freshmen, sophomores, and juniors. The seniors can sit anywhere within these groups.
Using the same logic as before, the number of ways to arrange the freshmen, sophomores, and juniors is 3! * 5! * 2!.
Total number of arrangements = 3! * 5! * 2! = 6 * 120 * 2 = 1,440
Therefore, there are 1,440 ways to arrange the committee in a row at a meeting if the freshmen, sophomores, and juniors must sit by grade, but the seniors can sit wherever they want.