In how many ways can a group of 10 people be divided into:

a. two groups consisting of 7 and 3 people?
b. three groups consisting of 4, 3, and 2 people?

This is a combination

Do you know that 5! = 5x4x3x2x1 =120??

from the 10 people you will choose 7

10! divided by (7!3!)

Once you have your group of 7 then there are only 3 people left to put into the group of 3. That is only 1 way.
I think the answer is 4320 ways.

for 3 groups.

10!/4!6! for the first group

You chose 4 so you have 6 left to choose from.

You have 3 left to choose from

3!/2!1!

You will multiply the values from these 3 combination calculations to get your final answer.

6!/3!3! for the second group.

a. Well, if you're asking about dividing 10 people into two groups consisting of 7 and 3 people, there's really only one way to do it. You take 7 people and put them in one group, and you take the other 3 people and put them in the other group. Voila! Division complete.

b. Now, if you want to divide those 10 people into three groups consisting of 4, 3, and 2 people, the possibilities are a bit more interesting. Let's see... you could have Group A with 4 people, Group B with 3 people, and Group C with 2 people. Or you could mix it up and have Group A with 2 people, Group B with 4 people, and Group C with 3 people. Or... well, you get the idea. There are actually a few different possibilities for this one, so let's just say there are multiple ways to divide those 10 people into three groups. Variety is the spice of life, after all!

a. To divide a group of 10 people into two groups consisting of 7 and 3 people, we need to calculate the number of combinations.

Step 1: Choose the group of 7 people:
The number of ways to choose 7 people out of 10 is given by the combination formula:
C(10, 7) = 10! / (7! * (10-7)!) = 10! / (7! * 3!) = (10 * 9 * 8) / (3 * 2 * 1) = 120.

Step 2: The remaining 3 people form the second group.
There is only one way to select the remaining 3 people.

Therefore, the total number of ways to divide the group of 10 people into two groups, with one consisting of 7 people and the other consisting of 3 people, is 120.

b. To divide a group of 10 people into three groups consisting of 4, 3, and 2 people, we need to calculate the number of combinations.

Step 1: Choose the group of 4 people:
The number of ways to choose 4 people out of 10 is given by the combination formula:
C(10, 4) = 10! / (4! * (10-4)!) = 10! / (4! * 6!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210.

Step 2: Choose the group of 3 people from the remaining 6 people:
The number of ways to choose 3 people out of 6 is given by the combination formula:
C(6, 3) = 6! / (3! * (6-3)!) = 6! / (3! * 3!) = (6 * 5 * 4) / (3 * 2 * 1) = 20.

Therefore, the total number of ways to divide the group of 10 people into three groups, with one consisting of 4 people, the second consisting of 3 people, and the third consisting of 2 people, is 210 * 20 = 4200.

To find the number of ways a group of 10 people can be divided into specific group sizes, we can use combinatorics.

a. To divide the group into two groups consisting of 7 and 3 people:

We need to choose 7 people from the group of 10 to form the first group, and the remaining 3 people automatically form the second group.

The number of ways to choose 7 people from a group of 10 can be calculated using the combination formula, also known as "nCr." The formula for nCr is:

nCr = n! / (r! * (n-r)!)

Where n is the total number of items and r is the number of items to be chosen.

In this case, n = 10 (total number of people) and r = 7 (number of people to be chosen for the first group).

10C7 = 10! / (7! * (10-7)!)
= 10! / (7! * 3!)
= (10 * 9 * 8 * 7!) / (7! * 3!)
= 10 * 9 * 8 / 3 * 2 * 1
= 120

Therefore, there are 120 ways to divide a group of 10 people into two groups consisting of 7 and 3 people.

b. To divide the group into three groups consisting of 4, 3, and 2 people:

We need to choose 4 people for the first group, 3 people for the second group, and the remaining 3 people automatically form the third group.

Similarly, we can use the combination formula to calculate the number of ways to choose the required number of people for each group.

10C4 * 6C3 = (10! / (4! * 6!)) * (6! / (3! * 3!))
= (10 * 9 * 8 * 7!) / (4! * 6 * 5 * 4!) * (6 * 5 * 4!) / (3! * 3 * 2 * 1)
= 210 * 20
= 4200

Therefore, there are 4200 ways to divide a group of 10 people into three groups consisting of 4, 3, and 2 people.