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A committee must be formed with 3 teachers and 6 students. If there are 8 teachers to choose from, and 17 students, how many different ways could the committee be made?
Answer:

A committee must be formed with $3$ teachers and $6$ students. If there are $8$ teachers to choose from, and $17$ 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

6

students. If there are

8

teachers to choose from, and

17

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 need to choose $3$ out of $8$ , which is a combination problem. The formula for combinations is $C(n, k) = \frac{n!}{k!(n-k)!}$ , where $n$ is the total number to choose from, $k$ is the number 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: $\newline$ $C(8, 3) = \frac{8!}{3!(8-3)!} = \frac{8!}{3!5!} = \frac{8\times7\times6}{3\times2\times1} = 56$ ways to choose the teachers.
Multiply Teachers and Students Combinations: Next, we calculate the number of ways to choose $6$ students out of $17$ . Again, using the combination formula: $\newline$ $C(17, 6) = \frac{17!}{(6!(17-6)!)} = \frac{17!}{(6!11!)} = \frac{(17\times16\times15\times14\times13\times12)}{(6\times5\times4\times3\times2\times1)} = 12376$ ways to choose the students.
Total Number of Committees: Now, we multiply the number of ways to choose the teachers by the number of ways to choose the 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 = $56 \times 12376$ .
Total Number of Committees: Now, we multiply the number of ways to choose the teachers by the number of ways to choose the 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 = $56 \times 12376$ .Performing the multiplication gives us the total number of different committees: $\newline$ Total number of committees = $56 \times 12376 = 693,376$ .

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