![]() □ ≥ □ ≥ 1, we have the following properties: Properties: Related Permutations (Part 2) There are also various other properties that relate similar permutations, which Thus, the solution set isĪs we saw in the previous example, the property of related permutations can beĮxtended further by applying it multiple times in succession. Out of these two solutions, note that only the positive one is valid, since We have factored this by noting that 12 and 19 are factors of 228,īut note that this can also be solved using the quadratic equation or byįinally, we can solve this equation by setting the factors to zero, giving us ![]() ![]() Then, dividing by □ on each side gives usĪs this is a quadratic equation, we can solve it for □ byĮxpanding the parentheses and rearranging everything to one side, before If we apply this formula to the right-hand side of the given equation, we getĢ 4 0 □ = □ = ( □ + 4 ) ( □ + 3 ) □. Us applying it multiple times in succession. Recall that we have the following property for relatingĪlthough this only applies to decreasing the indices of We can see that the indices on the left have both been decreased by 2Ĭompared to the right. □ on either side of the equation. AnswerĪs a starting point to finding the solution set, we can compare Property that can be applied to any □-permutations ofĮxample 3: Evaluating Permutations to Find the Values of Unknownsįind the solution set of the equation 2 4 0 □ = □ . So, we have shown that □ = 5 □ . Notice that if we take the 5 out of the numerator, we can get a different number Thus, we can consider what values ofĪnother technique we should be aware of is how we can relate permutations with That is, if □ = 0, then both sides of the equation will However, let us recall one of the properties of Since the values of □ are different for each side of theĮquation, it may not seem at first as though there is any way to make the The given equation is expressed in terms of partial permutations, which are Example 1: Using the Properties of Permutations to Find the Value of
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