In how many ways can a committee consisting of 2 faculty members and 3 students be for there are 11 faculty members and 14 students eligible to serve on the committee?
To find the number of ways to form a committee consisting of 2 faculty members and 3 students, we need to calculate the number of combinations.
The number of ways to choose 2 faculty members out of 11 can be calculated using the combination formula:
C(n, r) = n! / (r!(n-r)!)
In this case, n = 11 (number of faculty members) and r = 2 (number of slots for faculty members on the committee). Therefore,
C(11, 2) = 11! / (2!(11-2)!) = 11! / (2!9!)
Simplifying this expression, we get:
C(11, 2) = (11 * 10) / (2 * 1) = 55
Similarly, the number of ways to choose 3 students out of 14 is:
C(14, 3) = 14! / (3!(14-3)!) = 14! / (3! 11!)
Simplifying this expression, we get:
C(14, 3) = (14 * 13 * 12) / (3 * 2 * 1) = 364
Finally, to find the total number of ways to form the committee, we multiply the number of ways to choose faculty members by the number of ways to choose students:
Total number of ways = C(11, 2) * C(14, 3) = 55 * 364 = 20,020
Therefore, there are 20,020 ways to form the committee consisting of 2 faculty members and 3 students.
To find the number of ways to form a committee with 2 faculty members and 3 students from a pool of 11 faculty members and 14 students, we can use the concept of combinations.
The number of ways to choose 2 faculty members from 11 is given by the combination formula:
C(n, r) = n! / (r!(n-r)!)
where n is the total number of items, r is the number of items to be chosen, and ! denotes the factorial of a number.
In this case, we have n = 11 (faculty members) and r = 2 (faculty members to be chosen). Plugging these values into the combination formula:
C(11, 2) = 11! / (2! (11-2)!)
= 11! / (2! 9!)
Simplifying further:
C(11, 2) = (11*10*9!) / (2! 9!)
= (11*10) / 2!
= 55
Therefore, there are 55 ways to choose 2 faculty members from a pool of 11.
Similarly, the number of ways to choose 3 students from 14 is given by the combination formula:
C(14, 3) = 14! / (3! (14-3)!)
Again, plugging in the values:
C(14, 3) = 14! / (3! 11!)
Simplifying further:
C(14, 3) = (14*13*12*11!) / (3! 11!)
= (14*13*12) / 3!
= 286
There are 286 ways to choose 3 students from a pool of 14.
To find the total number of ways to form the committee, we need to multiply the number of ways to choose faculty members and the number of ways to choose students:
Total ways = C(11, 2) * C(14, 3)
= 55 * 286
= 15730
Therefore, there are 15,730 ways to form a committee consisting of 2 faculty members and 3 students from a pool of 11 faculty members and 14 students.