Examples of Counting Problems

By Steve Mussmann, proud Member of the Math Squad.


Here are some counting problems. A tutorial and solutions to these questions can be found here.


Q: How many ways are there to arrange the letters in the alphabet?

Q: An Olympic race has 100 contestants. How many ways can the gold, silver, and bronze medals be distributed?

Q: A wealthy car manufacturer is writing her will. She has 3 different cars and 100 grandchildren. She won't give multiple cars to one grandchild. How many ways are there to give the cars away in her will?

Q: A town security system has 10 security shifts available and 22 applicants. How many ways the manager hire applicants for the 10 different positions?

Q: The Omaha Zoo is hiring 12 zookeepers. They have 300 applicants. How many ways to hire 12 zookeepers from the applicants?

Q: In the United States, there are 50 states. Suppose a company wants to build 15 distribution centers around the US, 5 major centers and 10 minor centers. Further suppose that every state cannot have more than one center. How many ways to place the distribution centers?

Q: A landscaping company has 6 employees (as in the introduction). How many ways to divide the employees into two teams of three?

Q: A basketball camp has 30 players. The administration wishes to divide the players into 6 teams of 5 players. How many ways are there to do this?

Q: A organization is giving away 12 artistically designed (distinguishable) vases to its officers. It will be sending some to the marketing department, some to the research department, and some to the production department. Each department can have any number of vases. How many ways are there to divide the vases among the recipients?

Q: An art class has 20 students. The teacher has bought materials for 5 different art projects, one of which is a gold leaf project. The teacher doesn't mind if some of the projects have many or no students working on them except the teacher wants at least one student to work on the gold leaf project to prevent the gold from going to waste. How many ways are there for the students to be divided among the projects?


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The Spring 2013 Math Squad 2013 was supported by an anonymous gift to Project Rhea. If you enjoyed reading these tutorials, please help Rhea "help students learn" with a donation to this project. Your contribution is greatly appreciated.

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