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?
To find the number of ways to choose the committee, we need to consider the number of ways to choose members from each age group.
First, let's calculate the number of ways to choose 5 people from the 21-30 age group out of 6 people. This can be calculated using the combination formula, denoted as C(n, r), which represents the number of ways to choose r items from a set of n items without regard to the order.
The formula for combination is given by:
C(n, r) = n! / (r!(n-r)!)
Using this formula, we can calculate the number of ways to choose 5 people from 6:
C(6, 5) = 6! / (5!(6-5)!) = 6! / (5! × 1!) = 6
Next, we need to calculate the number of ways to choose 4 people from the 31-40 age group out of 6 people:
C(6, 4) = 6! / (4!(6-4)!) = 6! / (4! × 2!) = 15
Similarly, for the 41-50 age group, we need to choose 3 people out of 6:
C(6, 3) = 6! / (3!(6-3)!) = 6! / (3! × 3!) = 20
Now, we can multiply the number of ways to choose from each age group to get the total number of ways to choose the committee:
Total number of ways = Number of ways to choose from 21-30 age group × Number of ways to choose from 31-40 age group × Number of ways to choose from 41-50 age group
= 6 × 15 × 20
= 1800
Therefore, there are 1800 ways to choose the committee of 12 people 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 can use the concept of combinations. 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.
The number of ways to choose 5 from the 6 people in the 21-30 age group is denoted as C(6, 5) which is calculated as:
C(6, 5) = 6! / (5! * (6-5)!) = 6
Similarly, the number of ways to choose 4 from the 6 people in the 31-40 age group is C(6, 4), which is calculated as:
C(6, 4) = 6! / (4! * (6-4)!) = 15
Lastly, the number of ways to choose 3 from the 6 people in the 41-50 age group is C(6, 3), which is calculated as:
C(6, 3) = 6! / (3! * (6-3)!) = 20
To find the total number of ways to form the committee, we multiply the three values together, since the selection of people in each age group is independent:
Total number of ways = C(6, 5) * C(6, 4) * C(6, 3) = 6 * 15 * 20 = 1800
Therefore, there are 1800 ways to choose a committee of 12 people with the given conditions.