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A permutation is an arrangement of objects in a specific order. In a permutation, each object appears exactly once, and the order in which the objects are arranged matters. More formally, a permutation of the set \(A\) is a bijection, which is a one-to-one correspondence (mapping) from the set \(A\) to itself. The set \(A\) can be finite or infinite.

In this context, "order 3" means that we are considering the permutations involving three distinct elements. Let's consider a set of three distinct elements: \(\{a, b, c\}\). The aim is to find all possible permutations of these three elements. To do this, we can follow these steps: 1. Select an element to be the first element of the permutation. 2. Select an element from the remaining elements to be the second element of the permutation. 3. Place the last remaining element as the third element of the permutation. Let's go through all the possible permutations step by step: 1. Start with the first element as \(a\): - Second element can be either \(b\) or \(c\): - If second element is \(b\), the third element is \(c\). So, the permutation is: \(abc\). - If second element is \(c\), the third element is \(b\). So, the permutation is: \(acb\). 2. Next, consider the first element as \(b\): - Second element can be either \(a\) or \(c\): - If the second element is \(a\), the third element is \(c\). So, the permutation is: \(bac\). - If the second element is \(c\), the third element is \(a\). So, the permutation is: \(bca\). 3. Finally, let the first element be \(c\): - Second element can be either \(a\) or \(b\): - If the second element is \(a\), the third element is \(b\). So, the permutation is: \(cab\). - If the second element is \(b\), the third element is \(a\). So, the permutation is: \(cba\). So, the permutations of order 3 for the set \(\{a,b,c\}\) are: \(abc\), \(acb\), \(bac\), \(bca\), \(cab\), and \(cba\).