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A committee must be formed with teachers and students. If there are teachers to choose from, and students, how many different ways could the committee be made?Answer:
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Q. A committee must be formed with teachers and students. If there are teachers to choose from, and students, how many different ways could the committee be made?Answer:
- Calculate Teachers Combination: Calculate the number of ways to choose teachers out of . We use the combination formula, which is , where is the total number of items to choose from, is the number of items to choose, and ＂!＂ denotes factorial. For teachers, and . C(\(7\), \(4\)) = \frac{\(7\)!}{\(4\)!(\(7\)\(-4\))!} = \frac{\(7\)!}{\(4\)!\(3\)!} = \frac{(\(7\)\times\(6\)\times\(5\)\times\(4\)!)}{(\(4\)!\times\(3\)\times\(2\)\times\(1\))} = \frac{(\(7\)\times\(6\)\times\(5\))}{(\(3\)\times\(2\)\times\(1\))} = \(35\)
- Calculate Students Combination: Calculate the number of ways to choose \(4\) students out of \(14\). Again, we use the combination formula. For students, \(n = 14\) and \(k = 4\). C(, ) = \frac{!}{!()!} = \frac{!}{!!} = \frac{\times\times\times\times!}{!\times!} = \frac{\times\times\times}{\times\times\times} = \frac{\times\times\times}{} =
- Calculate Total Number of Ways: Calculate the total number of ways to form the committee by multiplying the number of ways to choose teachers by the number of ways to choose students.Total number of ways Number of ways to choose teachers Number of ways to choose students
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