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Suppose we want to choose objects, without replacement, from distinct objects.(a) How many ways can this be done, if the order of the choices does not matter?(b) How many ways can this be done, if the order of the choices matters?
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Q. Suppose we want to choose objects, without replacement, from distinct objects.(a) How many ways can this be done, if the order of the choices does not matter?(b) How many ways can this be done, if the order of the choices matters?
- Calculate combinations: Calculate the number of ways to choose objects from without considering the order (combinations).Reasoning: Use the combination formula , where is the total number of objects and is the number of objects to choose.Calculation: \(15\)C\(5\) = \frac{\(15\)!}{\(5\)!(\(15\)\(-5\))!} = \frac{\(15\)!}{(\(5\)! \cdot \(10\)!)} = \frac{(\(15\) \cdot \(14\) \cdot \(13\) \cdot \(12\) \cdot \(11\))}{(\(5\) \cdot \(4\) \cdot \(3\) \cdot \(2\) \cdot \(1\))} = \(3003\).
- Calculate permutations: Calculate the number of ways to choose \(5\) objects from \(15\) considering the order (permutations).\(\newline\)Reasoning: Use the permutation formula \(nPr = \frac{n!}{(n-r)!}\), where \(n\) is the total number of objects and \(r\) is the number of objects to choose.\(\newline\)Calculation: P = \frac{!}{()!} = \frac{!}{!} = ( \times \times \times \times ) = .
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