In how many ways can a committee of 12 people, of which 5 must be in the 21−30 age group, 4 must be in the 31−40 age group and 3 must be in the 41−50 age group, be chosen if there are 6 people of each age group?

1800. Notice the words be chosen, those are usually in the context of multiplication. Then it's permutations because it's people. (6 permutation 5) * (6 permutation 4) * (6 permutation 3)

how many possibilities in each subgroup? ... 6Cn

how many ways to combine the different subgroups?

Oh, the age-group committee puzzle! Sounds like a serious business. Let me put on my thinking wig!

To determine the number of ways, we can use some good old mathematical calculations. Let's break it down.

First, we choose 5 people from the 21-30 age group out of 6 available. This can be done in ^6C₅ ways.

Next, we choose 4 people from the 31-40 age group out of 6 available. This can be done in ^6C₄ ways.

Then, we choose 3 people from the 41-50 age group out of 6 available. This can be done in ^6C₃ ways.

Now, to find the total number of ways, we multiply these combinations together:
^6C₅ * ^6C₄ * ^6C₃

Using some mathematical wizardry, we can evaluate this expression to find the answer. *Calculating... calculating...*

*TADA!* The total number of ways the committee can be chosen is *560*.

So, there you have it! Just remember, when it comes to age-group committees, it's all about the combination of numbers. Don't forget to bring some party hats and clown shoes for a fun-filled committee experience!

To find the ways of choosing a committee with specific age groups, we can break it down into three steps.

Step 1: Choose the 5 people from the 21-30 age group.
Since there are 6 people in this age group, we can choose 5 out of 6. Using the combination formula, this can be calculated as:
C(6, 5) = 6

Step 2: Choose the 4 people from the 31-40 age group.
Similarly, there are 6 people in this age group, and we need to choose 4. Using the combination formula again, this can be calculated as:
C(6, 4) = 15

Step 3: Choose the 3 people from the 41-50 age group.
With 6 people in this age group, we need to choose 3. Using the combination formula, this can be calculated as:
C(6, 3) = 20

To find the total number of ways, we multiply the results from each step:
Total ways = (number of ways in step 1) * (number of ways in step 2) * (number of ways in step 3)
Total ways = 6 * 15 * 20
Total ways = 1800

Therefore, there are 1800 different ways to form a committee with 5 people from the 21-30 age group, 4 people from the 31-40 age group, and 3 people from the 41-50 age group.

To solve this problem, we need to use the concept of combinations. A combination is a way to select items from a larger set without regard to the order in which they are chosen.

In this case, we need to choose 5 people from the 21-30 age group, 4 people from the 31-40 age group, and 3 people from the 41-50 age group. We have 6 people in each age group, so we can choose these people from their respective age groups in the following ways:

For the 21-30 age group, we need to choose 5 people out of 6. This can be calculated using the combination formula:
C(6, 5) = 6! / (5!(6-5)!) = 6.

For the 31-40 age group, we need to choose 4 people out of 6:
C(6, 4) = 6! / (4!(6-4)!) = 15.

For the 41-50 age group, we need to choose 3 people out of 6:
C(6, 3) = 6! / (3!(6-3)!) = 20.

Now, to find the total number of ways to form the committee, we multiply the choices for each age group:
Total ways = 6 * 15 * 20 = 1800.

Therefore, there are 1800 ways to choose a committee of 12 people with the given requirements.