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Chapter 33: Chapter 33

Problem 1056

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Find \(_{9} P_{4}\)

Short Answer

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The number of ways to arrange 4 objects out of a total of 9 objects, represented as \(_{9}P_{4}\), is 3024.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Understanding the Permutation Formula

The general formula for a permutation of n objects taken r at a time is denoted \(_{n}P_{r}\) and is given by: \[ _{n}P_{r} = \frac{n!}{(n-r)!}, \] where n! (n factorial) denotes the product of all positive integers up to n. Now, we will apply this formula to our specific problem to find \(_{9}P_{4}\).

Step 2: Applying the Permutation Formula

We are given n = 9 and r = 4. Plugging these values into the Permutation Formula, we obtain: \[ _{9}P_{4} = \frac{9!}{(9-4)!}. \]

Step 3: Computing Factorials and Simplifying

Now we will compute the factorials and simplify the expression: \[ _{9}P_{4} = \frac{9!}{5!} = \frac{9\times8\times7\times6\times5\times4\times3\times2}{5\times4\times3\times2} \\ \] Notice how the 5, 4, 3, and 2 terms cancel out, leaving us with: \[ _{9}P_{4} = 9\times8\times7\times6 \]

Step 4: Multiplying the Permutation Result

Now, we will multiply the remaining numbers to get the final permutation value: \[ _{9}P_{4} = 9\times8\times7\times6 = 3024 \]

Step 5: The Final Answer

The final result for the permutation \(_{9}P_{4}\) is 3024. This means that there are 3024 different ways to arrange 4 out of 9 objects in an ordered manner.

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