Twelve children are divided up into three groups, of five, four, and three children, respectively. In how many ways can this be done if the order within each group is not important?

Thank you so much!

Well if you divide 12 children into 5 groups you might find yourself in jail....I'm just saying

What you are asking is how many ways to arrange 3 different numbers. In general, if you have n items, there are n! ways to arrange them (permutations).

3! = 3*2*1

3 4 5
3 5 4
4 3 5
4 5 3
5 4 3
5 3 4

There are 3 ways to pick the first group
That leaves only 2 ways to pick the next group,
and only 1 way left to pick the 3rd group.

3*2*1 = 6 = 3!

To solve this problem, we can use a technique called combinations. In this case, we want to find the number of ways to divide twelve children into three groups: one group of five children, one group of four children, and one group of three children.

First, let's consider the group of five children. We need to choose five children out of the twelve available. We can calculate this using a combination:

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

where C(n, k) represents the number of ways to choose k objects from a set of n objects. In this case, n is 12 (the total number of children) and k is 5 (the number of children in the first group).

Using the combination formula, we can calculate the number of ways to select five children from twelve:

C(12, 5) = 12! / (5!(12-5)!) = 12! / (5! * 7!) = (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1) = 792.

So, there are 792 ways to select five children out of twelve for the first group.

Next, let's consider the group of four children. We need to choose four children out of the remaining seven children. Again, we can use the combination formula:

C(7, 4) = 7! / (4!(7-4)!) = 7! / (4! * 3!) = (7 * 6 * 5) / (3 * 2 * 1) = 35.

So, there are 35 ways to select four children out of the remaining seven for the second group.

Finally, we have three children left for the last group, so there is only one way to assign them to the third group.

Now, to find the total number of ways to divide the twelve children into three groups, we need to multiply the number of ways for each group together:

Total number of ways = Number of ways for the first group * Number of ways for the second group * Number of ways for the third group
= 792 * 35 * 1
= 27720.

Therefore, there are 27,720 ways to divide twelve children into three groups of five, four, and three children, respectively, where the order within each group is not important.