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A committee must be formed with 5 teachers and 5 students. If there are 9 teachers to choose from, and 14 students, how many different ways could the committee be made?
Answer:

A committee must be formed with $5$ teachers and $5$ students. If there are $9$ teachers to choose from, and $14$ 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

5

teachers and

5

students. If there are

9

teachers to choose from, and

14

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

\newline

Answer:

Calculate Combinations for Teachers: 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 Combinations for Students: First, we calculate the number of ways to choose $5$ teachers out of $9$ . Using the combination formula: $\newline$ $C(9, 5) = \frac{9!}{5!(9 - 5)!} = \frac{9!}{5!4!} = \frac{9\times8\times7\times6\times5!}{5!\times4\times3\times2\times1} = \frac{9\times8\times7\times6}{4\times3\times2\times1} = 126$
Multiply Combinations: Next, we calculate the number of ways to choose $5$ students out of $14$ . Using the combination formula again: $\newline$ $C(14, 5) = \frac{14!}{(5!(14 - 5)!)} = \frac{14!}{(5!9!)} = \frac{(14\times13\times12\times11\times10\times9!)}{(5!\times9!)} = \frac{(14\times13\times12\times11\times10)}{(5\times4\times3\times2\times1)} = 2002$
Calculate Total Ways: Now, we multiply the number of combinations of teachers by the number of combinations of students to find the total number of ways to form the committee: Total number of ways = Number of ways to choose teachers $\times$ Number of ways to choose students = $126 \times 2002$
Perform Multiplication: Perform the multiplication to find the total number of ways: $\newline$ Total number of ways = $126 \times 2002 = 252252$

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