In: Statistics and Probability
A committee of four experts is being assembled to tackle a
recent outbreak of a virus. The government has provided a list of
10 virologists, 3 epidemiologists and 6 mathematicians who are
suitable to be on the committee. How many committees are possible
if the members are chosen at random?
How many committees are possible if each group is represented by at
least one committee member?
The government has provided a list of 10 virologists, 3 epidemiologists and 6 mathematicians
A committee of four experts is being assembled
Question (1)
Number of committees that can be formed if the members are chosen at random are selecting 4 experets from total 19 experts which can be done in ways
= 19! / [ 4 ! * (19 -4) ! ]
= 3876 committees
So 3876 committees are possible if the members are chosen at random
Question (2)
Number of committees that are possible if each group is represented by at least one committee member
Here there should be atleast one committee members from each group. So three of the four members in the committee are from each of the groups
Case 1
The 4th member is from virologists which implies there will be 2 virologists, 1 epidemiologist and 1 mathematician in the committee
The number of committees in this case = * *
= 45 * 3 * 6
= 810
Case 2
The 4th member is from epidemiologists which implies there will be 1 virologist, 2 epidemiologist and 1 mathematician in the committee
The number of committees in this case = * *
= 10 * 3 * 6
= 180
Case 3
The 4th member is from mathematicians which implies there will be 1 virologist, 1 epidemiologist and 2 mathematicians in the committee
The number of committees in this case = * *
= 10 * 3 * 15
= 450
So total number of committees that can be formed if each group is represented by at least one committee member = 810 + 180 + 450 = 1440 committees
So Answer is 1440 committees