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 form the committee with the given conditions, we need to use the concept of combinations.
First, let's calculate the number of ways to choose the 5 people from the 21-30 age group. Since there are 6 people in this age group, it can be done using combinations:
C(6, 5) = 6! / (5!(6-5)!) = 6
Next, let's calculate the number of ways to choose the 4 people from the 31-40 age group:
C(6, 4) = 6! / (4!(6-4)!) = 15
Finally, let's calculate the number of ways to choose the 3 people from the 41-50 age group:
C(6, 3) = 6! / (3!(6-3)!) = 20
To find the total number of ways to form the committee satisfying the given conditions, we need to multiply these three numbers:
Total ways = 6 * 15 * 20 = 1800
Therefore, there are 1800 ways to choose the committee of 12 people, with 5 from the 21-30 age group, 4 from the 31-40 age group, and 3 from the 41-50 age group, given that there are 6 people in each age group.