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A committee must be formed with 3 teachers and 3 students. If there are 8 teachers to choose from, and 16 students, how many different ways could the committee be made?
Answer:

A committee must be formed with $3$ teachers and $3$ students. If there are $8$ teachers to choose from, and $16$ students, how many different ways could the committee be made? $\newline$ Answer:

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Counting principle

Full solution

Q. A committee must be formed with

3

teachers and

3

students. If there are

8

teachers to choose from, and

16

students, how many different ways could the committee be made?

\newline

Answer:

Calculate Teachers Combinations: To determine the number of different ways to form the committee, we need to calculate the combinations of teachers and students separately and then multiply them together. For the teachers, we will use the combination formula which is $C(n, k) = \frac{n!}{k!(n - k)!}$ , where $n$ is the total number of items to choose from, $k$ is the number of items to choose, and $＂!＂$ denotes factorial.
Calculate Students Combinations: First, we calculate the number of ways to choose $3$ teachers out of $8$ . Using the combination formula: $\newline$ $C(8, 3) = \frac{8!}{3!(8 - 3)!} = \frac{8!}{3!5!} = \frac{(8 \times 7 \times 6)}{(3 \times 2 \times 1)} = 56$ ways to choose the teachers.
Multiply Teachers and Students Combinations: Next, we calculate the number of ways to choose $3$ students out of $16$ . Using the combination formula again: $\newline$ $C(16, 3) = \frac{16!}{(3!(16 - 3)!)} = \frac{16!}{(3!13!)} = \frac{(16 \times 15 \times 14)}{(3 \times 2 \times 1)} = 560$ ways to choose the students.
Find Total Number of Committees: Now, we multiply the number of ways to choose teachers by the number of ways to choose students to find the total number of different committees that can be formed. $\newline$ Total number of committees = Number of ways to choose teachers $\times$ Number of ways to choose students = $56 \times 560$ .
Perform Multiplication: Perform the multiplication to find the total number of different committees: Total number of committees = $56 \times 560 = 31,360$ .

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