6 teachers and 12 students volunteer for a committee to discuss extra curricular activities. How many committees of 5 people can be made if barry and miss morrison cannot be on the committee togetehr
Since Barry and Miss Morrison cannot be on the committee together, we have to consider two cases: one where Barry is on the committee and Miss Morrison is not, and one where Miss Morrison is on the committee and Barry is not.
Case 1: Barry is on the committee and Miss Morrison is not.
In this case, we have to select 4 more people from the remaining 6 teachers and 12 students, excluding Miss Morrison. The number of ways to choose 4 people from 6 teachers and 12 students is given by the combination formula: C(6+12-1, 4) = C(17, 4) = 2380.
Case 2: Miss Morrison is on the committee and Barry is not.
Similarly, in this case, we have to select 4 more people from the remaining 6 teachers and 12 students, excluding Barry. The number of ways to choose 4 people from 6 teachers and 12 students is again given by the combination formula: C(6+12-1, 4) = C(17, 4) = 2380.
Thus, there are a total of 2380 + 2380 = 4760 committees of 5 people that can be made if Barry and Miss Morrison cannot be on the committee together.
To solve this problem, we need to use a combination formula.
The total number of teachers available for the committee is 6.
The total number of students available for the committee is 12.
The committee needs to consist of 5 people, and Barry and Miss Morrison cannot be on the committee together.
We will find the total number of committees by subtracting the number of committees that include both Barry and Miss Morrison from the total number of committees.
Step 1: Calculate the total number of committees that include both Barry and Miss Morrison.
Number of committees including both Barry and Miss Morrison = 1 (because they both need to be in the committee) × C(5-2, 3) (choosing 3 more people from the remaining 5-2 = 3 people)
C(5-2, 3) = C(3, 3) = 1 (choosing all 3 remaining people in a committee of 3)
Number of committees including both Barry and Miss Morrison = 1
Step 2: Calculate the total number of committees without both Barry and Miss Morrison.
Number of committees without both Barry and Miss Morrison = C(6, 5) (choosing 5 people from the 6 teachers) × C(12, 0) (choosing 0 students from the 12 students)
C(6, 5) = 6 (choosing 5 teachers from 6 teachers)
C(12, 0) = 1 (choosing 0 students from 12 students)
Number of committees without both Barry and Miss Morrison = 6 × 1 = 6
Step 3: Calculate the total number of committees by subtracting the number of committees that include both Barry and Miss Morrison from the total number of committees without both Barry and Miss Morrison.
Total number of committees = Number of committees without both Barry and Miss Morrison - Number of committees including both Barry and Miss Morrison
Total number of committees = 6 - 1 = 5
Therefore, there are 5 committees of 5 people that can be made without Barry and Miss Morrison being together.