Suppose that a state university has to form a committee of 5 members from a list of 20 candidates out of whom 12 are teachers and 8 are students.If members of the committee are selected at random,what is the probability that the majority of the committee members are students?

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suppose that a state university has to form a committee of 5 members from a list of 20 candidates out of whom 12 are teachers and 8 are student. if the member of the committee are selected at redom, what is the probability that the majority of the committee member are student?

To find the probability that the majority of the committee members are students, we need to determine the number of successful outcomes (committee compositions with the majority of students) and divide it by the total number of possible outcomes.

Number of successful outcomes:
To have the majority of the committee members as students, we need at least 3 out of the 5 committee members to be students.
There are 8 students in the candidate list, so we have to calculate the combinations of 3, 4, and 5 students in a committee of 5 members:
1. Select 3 students from 8: C(8,3)
2. Select 4 students from 8: C(8,4)
3. Select all 5 students from 8: C(8,5)

Number of total outcomes:
To calculate the total number of outcomes, we need to consider all possible ways of selecting a committee of 5 members from the list of 20 candidates.
There are 20 candidates in total, so we have to calculate the combinations of selecting 5 members from 20: C(20,5)

Probability:
Finally, we can calculate the probability by dividing the number of successful outcomes by the number of total outcomes:

Probability = (Number of successful outcomes) / (Number of total outcomes)
= (C(8,3) + C(8,4) + C(8,5)) / C(20,5)

We can use these values to calculate the probability.

To find the probability that the majority of the committee members are students, we need to determine the number of ways the majority can be formed and divide it by the total number of possible committees.

First, let's calculate the total number of possible committees. From a group of 20 candidates, we need to choose 5 members. This can be done using the combination formula:

C(n, r) = n! / (r!(n-r)!)

Where n is the total number of candidates, and r is the number of committee members to be chosen.

In this case, n = 20 and r = 5. Plugging these values into the formula:

C(20, 5) = 20! / (5!(20-5)!) = 20! / (5!15!) = (20*19*18*17*16) / (5*4*3*2*1) = 15,504

So there are 15,504 different possible committees of 5 members that can be formed from the 20 candidates.

Next, let's calculate the number of committees where the majority are students.

Since there are 8 student candidates and 12 teacher candidates, there are 3 possibilities for forming the majority:

1. Three students and two teachers
2. Four students and one teacher
3. Five students (which is the maximum majority possible)

For each of these possibilities, we need to calculate the number of ways it can occur. We can use the combination formula again.

1. Three students and two teachers:
C(8, 3) * C(12, 2) = (8! / (3!(8-3)!)) * (12! / (2!(12-2)!)) = 56 * 66 = 3,696

2. Four students and one teacher:
C(8, 4) * C(12, 1) = (8! / (4!(8-4)!)) * (12! / (1!(12-1)!)) = 70 * 12 = 840

3. Five students:
C(8, 5) = 8! / (5!(8-5)!) = 56

Now, we can sum up the number of committees with the majority as students:

3,696 (three students and two teachers) + 840 (four students and one teacher) + 56 (five students) = 4,592

Finally, we can calculate the probability by dividing the number of committees with the majority as students by the total number of possible committees:

P(majority as students) = 4,592 / 15,504 ≈ 0.2969

Therefore, the probability that the majority of the committee members are students is approximately 0.2969 or 29.69%.