Logic

Derangements

You are examining a vegetable storage silo and you come upon a room with six different bins. Each bin contains exactly 1 unique type of vegetable from the following list: onions, carrots, potatoes, squash, celery, bell peppers.

Each bin is labeled with the name of a specific type of vegetable, such that all six types appear on exactly one label. However, further examination reveals that the bin labeled "carrots" contains potatoes. The one labelled "bell peppers" has onions, and the container labelled "celery" has squash.

How many bins are labelled correctly?

You have a set of three boxes, one filled with copper coins, one filled with silver coins, and one filled with gold coins. You decide to label the boxes according to their contents.

Unfortunately, your friend knocks the labels off, then reattaches them at random without checking to see if the labels are correct. When you discover this, you look inside the box labelled "Gold" to find that it contains silver coins.

What is the probability that your friend labeled at least 1 box correctly?

Joe is in a room with ten containers. Each container holds a unique item. Each container also has a unique label describing one of the items in the containers. However, each label has been incorrectly applied such that the label on each container does not match its contents.

Joe can currently see all the labels, but has no information about the contents of the containers. What is the minimum number of containers that Joe could open before it is possible that he could say with certainty that every container is mislabeled?

A teacher has a class of 4 students who have recently taken an exam. The teacher wants to have the students correct each other’s exams, but he doesn’t want any student to be correcting his or her own exam. If the teacher gives each student a completed exam at random, what is the probability that no student ends up with his or her own exam?

Joanna, Amy, Mike, Richard, and Cindy are participating in a gift exchange. Everyone will bring exactly 1 gift and leave with exactly 1 gift. Furthermore, no one is allowed to leave with the gift he or she brought.

If we know that Richard went home with Amy's gift and Joanna went home with Mike's gift, how many distinct ways are there to distribute the remaining gifts?

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