If there are 24 students in a class, how many different-sized groups can be formed, and how many students will be in each group? Make a list of all possible groupings.

2 and 12

3 and 8
4 and 6

You can also include

6 and 4
8 and 3
12 and 2

This last one is 12 students in two groups.

To find the number of different-sized groups that can be formed with 24 students, we need to determine all the divisors of 24. The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. Now, I will list all the possible groupings:

1 student in a group: 1 group of 24 students.
2 students in a group: 12 groups of 2 students each (1-2, 3-4, 5-6, 7-8, 9-10, 11-12, 13-14, 15-16, 17-18, 19-20, 21-22, 23-24).
3 students in a group: 8 groups of 3 students each (1-3-5, 2-4-6, 7-9-11, 8-10-12, 13-15-17, 14-16-18, 19-21-23, 20-22-24).
4 students in a group: 6 groups of 4 students each (1-4-7-10, 2-5-8-11, 3-6-9-12, 13-16-19-22, 14-17-20-23, 15-18-21-24).
6 students in a group: 4 groups of 6 students each (1-6-11-16-21-24, 2-7-12-17-22-3, 4-9-14-19-23).
8 students in a group: 3 groups of 8 students each (1-8-15-22, 2-9-16-23, 3-10-17-24).
12 students in a group: 2 groups of 12 students each (1-12-23, 2-13-24).
24 students in a group: 1 group of all 24 students.

So, in total, there are 1 + 12 + 8 + 6 + 4 + 3 + 2 + 1 = 37 different-sized groups that can be formed with 24 students, and the number of students in each group ranges from 1 to 24.

To determine the number of different-sized groups that can be formed and the number of students in each group, we can use a combination formula.

The formula for finding the number of combinations of k elements from a set of n elements is given by:

C(n, k) = n! / (k!(n-k)!)

In this case, we have 24 students, and we want to find all possible groupings. We can start by determining the number of different-sized groups. We'll consider groups ranging in size from 2 to 24.

First, let's calculate the number of groups for each group size:

For group size 2, we have C(24, 2) = 24! / (2!(24-2)!) = 24! / (2!22!) = (24 * 23) / (2) = 276
For group size 3, we have C(24, 3) = 24! / (3!(24-3)!) = 24! / (3!21!) = (24 * 23 * 22) / (3 * 2) = 24 * 23 = 552
For group size 4, we have C(24, 4) = 24! / (4!(24-4)!) = 24! / (4!20!) = (24 * 23 * 22 * 21) / (4 * 3 * 2) = 6,135
...
For group size 24, we have C(24, 24) = 24! / (24!(24-24)!) = 24! / (24!0!) = 1

We can continue this process up to group size 24.

Now, let's list all the possible groupings:

Group size 2: (1, 2), (1, 3), ..., (1, 24), (2, 3), (2, 4), ..., (23, 24)

Group size 3: (1, 2, 3), (1, 2, 4), ..., (22, 23, 24)

Group size 4: (1, 2, 3, 4), (1, 2, 3, 5), ..., (21, 22, 23, 24)

...

Group size 24: (1, 2, 3, ..., 24)

In total, there are a total of 538,257 possible groupings considering different-sized groups ranging from 2 to 24 students.