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Chapter 33: Chapter 33

Problem 1058

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Calculate the number of permutations of the letters \(a, b, c, d\) taken two at a time.

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There are 12 possible permutations of the letters \(a, b, c,\) and \(d\) taken two at a time.
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TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Identify values of n and r

In this exercise, we have the letters \(a, b, c, d\). So, the total number of objects (letters) is \(n = 4\). We want to find permutations taken two at a time, so \(r = 2\).

Step 2: Find Factorials

Now, we need the factorials of \(n=4\) and \((n-r) = (4-2) = 2\). 1. For \(4!\), we have: \(4! = 4\times 3\times 2\times 1 = 24\) 2. For \(2!\), we have: \(2! = 2\times 1 = 2\)

Step 3: Apply the Permutation Formula

Now, we will apply the formula to find the number of permutations: \(P(n, r) = \frac{n!}{(n-r)!}\) Plugging in the values, we get: \(P(4, 2) = \frac{4!}{2!} = \frac{24}{2} = 12\) So there are 12 possible permutations of the letters \(a, b, c,\) and \(d\) taken two at a time.

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