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Permutations with Repetition

Using powers when selecting r from a group of n

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Permutations with Repetition
Teacher Contributed

License Plates and Permutations


License Plates and Permutations


  • Permutations
  • Permutations with Repetition
  • Combinations

Student Exploration

How many license plates are possible in the U.K.?

License plates, also known as vehicle registration plates, are used for authorizes to be able to track vehicles and determine who the owner of a vehicle is. Whenever you purchase a car you have to register it and order plates for your vehicle. And then, whenever the police need to give a person a parking or speeding ticket, they use the plates number to enter the ticket into the police system.

In the United Kingdom (U.K.) the license plates are made up of the regional flag followed by a two-digit local area code, a two-digit age identifier (corresponding to the year the vehicle is registered), followed by a three-digit sequence of letters. See the image below for a visual, from http://en.wikipedia.org/wiki/Vehicle_registration_plates_of_the_United_Kingdom#Current_system .

1. Given the constraints stated above, how many different license plates are possible in the U.K.?

Extension Investigation

2. Why is this a Permutation problem, rather than a Combination problem? Explain.
3. Why is this a Permutation with Repetitions problem? Explain.
4. Need a little help? Here are some steps to help you.
a. Let’s say that the year is set. Calculate how many of the random three-letter sequences are possible at the end of the license plate.
b. Next, calculate how many two-digit year identifiers are possible.
c. Next, calculate how many two-letter area codes are possible.
d. Now, calculate how many of the two-letter area codes, followed by two-digit year identifiers, followed by the random three-letter sequences are possible.
5. How would you approach this problem differently if letters and digits could not be repeated? Explain.
6. Research what the license plates (or vehicle registration plates) requirements are in your state or country. Then calculate how many different plates are possible in your state or country based on the license plate set up.

Resources Cited


Connections to other CK-12 Subject Areas

  • Probability and Permutations
  • Permutations and Combinations Compared
  • Permutation Problems
  • Combinations.

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