From 5 statisticians and 6 economists, a committee consisting of 3 statisticians and 2 economists is to be formed. How many different committees can be formed if:
a. two particular statisticians must be in the committee?
b. one particular economist cannot be in the committee?
To solve these problems, we can use the concept of combinations, which represents the number of ways to select a certain number of items from a larger set without considering the order.
a. In this case, two particular statisticians must be in the committee. We need to choose 1 more statistician from the remaining 3 statisticians and choose 2 economists from the initial 6 economists.
The number of ways to choose 1 statistician from 3 is denoted as C(3, 1), which can be calculated as:
C(3, 1) = 3! / (1! * (3-1)!) = 3
The number of ways to choose 2 economists from 6 is denoted as C(6, 2), which can be calculated as:
C(6, 2) = 6! / (2! * (6-2)!) = 15
To obtain the total number of committees, we need to multiply these two values:
Total number of committees = C(3, 1) * C(6, 2) = 3 * 15 = 45
Therefore, there are 45 different committees that can be formed if two particular statisticians must be in the committee.
b. In this case, one particular economist cannot be in the committee. We need to select 3 statisticians from the initial 5 statisticians and choose 2 economists from the initial 6 economists, excluding the one particular economist who cannot be in the committee.
The number of ways to choose 3 statisticians from 5 is denoted as C(5, 3), which can be calculated as:
C(5, 3) = 5! / (3! * (5-3)!) = 10
The number of ways to choose 2 economists from the remaining 5 economists is denoted as C(5, 2), which can be calculated as:
C(5, 2) = 5! / (2! * (5-2)!) = 10
To obtain the total number of committees, we need to multiply these two values:
Total number of committees = C(5, 3) * C(5, 2) = 10 * 10 = 100
Therefore, there are 100 different committees that can be formed if one particular economist cannot be in the committee.
In summary:
a. Two particular statisticians must be in the committee: 45 different committees can be formed.
b. One particular economist cannot be in the committee: 100 different committees can be formed.