oranges. Permutations? Consider three covered bins containing apples, oranges, and a mixture of apples and oranges, respectively. The three bins have labels Apples, Oranges, and Apples/Oranges, but the labels have been moved so that all the labels are wrong. We are allowed to reach into one bin and select one piece of fruit (without seeing the rest). Prove that by selecting the right bin to sample, we can determine the correct labeling of the bins. Explain how this relates to permutations.
Take fruit first from the Apples/Oranges because all labels are incorrect, this bin cannot contain both apples and oranges.
1. If we get an apple, then this bin must be the apples bin. The Apples labeled bin is incorrect, so the Apples bin must be for oranges. Finally, the "Oranges" bin must contain both Apples/Oranges.
2. If we get an orange, then this bin must be the oranges bin. Since the Oranges label is incorrect, so the bin must be for Apples. Finally, the Apples/Oranges bin.
Take fruit first from the Apples/Oranges because all labels are incorrect, this bin cannot contain both apples and oranges.
1. If we get an apple, then this bin must be the apples bin. The Apples labeled bin is incorrect, so the Apples bin must be for oranges. Finally, the "Oranges" bin must contain both Apples/Oranges.
2. If we get an orange, then this bin must be the oranges bin. Since the Oranges label is incorrect, so the bin must be for Apples. Finally, the Apples/Oranges bin.