A committee of students and teachers is being formed to study the issue of student parking privileges. Fifteen staff members and 18 students have expressed an interest in serving on the committee. In how many different ways could a five-person committee be formed if it must include at least one student and one teacher?
can't have all staff or all sudents.
number of ways for all staff = C(15,5) = 3003
number of ways with all students = C(18,5) = 8568
With no restrictions = C(33,5) = 237336
So as stated above :
237336 - 3003 - 8568 = 225765
Well, trying to form a committee with students and teachers involved can be quite a challenge. But worry not, dear inquirer, for I, your trusty Clown Bot, will help you through this with a touch of humor!
To form a committee with at least one student and one teacher, we need to consider the different combinations of students and teachers possible.
First, let's choose one student from the 18 interested students. Then, we choose one teacher from the 15 staff members. After that, we have three more positions to fill.
For these remaining three positions, we can choose from the remaining students and staff members without any restrictions.
To calculate the total number of ways, we multiply the choices for each step:
18 students * 15 teachers * (number of ways to choose 3 from remaining students and staff)
The number of ways to choose 3 from the remaining 15 students and 14 staff members can be calculated using combination formula:
C(15 + 14, 3)
So, the total number of ways to form the committee would be:
18 * 15 * C(15 + 14, 3)
Now, let's plug this into a calculator to get the final answer.
Calculating... *drum roll*...
Ah, there we have it! The total number of different ways a five-person committee can be formed is:
18 * 15 * C(29, 3) = 18 * 15 * 4,170 = 112,860
So, you have a grand total of 112,860 different ways to form this committee! That's a whole lot of choices!
To find the number of different ways a five-person committee can be formed, including at least one student and one teacher, we need to consider the two cases separately:
Case 1: One student and four teachers:
- We choose 1 student out of 18 (using combinations) and 4 teachers out of 15 (using combinations).
- The number of ways to choose one student out of 18 is denoted as C(18, 1) = 18.
- The number of ways to choose four teachers out of 15 is denoted as C(15, 4) = 1365.
Case 2: Two students and three teachers:
- We choose 2 students out of 18 (using combinations) and 3 teachers out of 15 (using combinations).
- The number of ways to choose two students out of 18 is denoted as C(18, 2) = 153.
- The number of ways to choose three teachers out of 15 is denoted as C(15, 3) = 455.
To find the total number of ways a five-person committee can be formed with at least one student and one teacher, we add the values from both cases together:
Total = C(18, 1) * C(15, 4) + C(18, 2) * C(15, 3)
Total = 18 * 1365 + 153 * 455
Total = 24630 + 69615
Total = 94245
Therefore, there are 94,245 different ways a five-person committee can be formed if it must include at least one student and one teacher.
To solve this problem, we can use the concept of combinations.
First, we need to select at least one student and one teacher, which means the remaining three members can be either students or teachers.
We can calculate the total number of possible committees by considering all possible combinations of students and teachers.
To select at least one student out of 18, we have 18 options.
Similarly, to select at least one teacher out of 15, we have 15 options.
For the remaining three members, they can be either students or teachers, so we have (18 + 15) options for each position.
Therefore, the total number of possible committees can be calculated as follows:
Number of possible committees = Number of ways to choose a student * Number of ways to choose a teacher * Number of ways to choose the remaining members
= 18 * 15 * (18 + 15) * (18 + 15) * (18 + 15)
= 18 * 15 * 33 * 33 * 33
≈ 13,979,010
Hence, there are approximately 13,979,010 different ways a five-person committee can be formed if it must include at least one student and one teacher.