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Permutations with Repetition

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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.?
In order to determine the total number of different license plate we need to break it down to each part of the license plate. The EU country identifier is already determined as we are analyzing the possibilities in the U.K. So, we need to break it down into the (1) area code, (2) age identifiers, and (3) random letters.
Area code: This is a Permutation of two letters with repetition. \frac{26 \cdot 26}{2!}  =  \frac{676}{2}=338.
Age identifier: This is a Permutation of two numbers with repetition. \frac{10 \cdot 10}{2!}  = \frac{100}{2}=50.
Random letters: This is a Permutation of three letters with repetition. \frac{26 \cdot 26 \cdot 26}{3!}  =  \frac{17576}{6}=2929.
Total number of different license plates: 338 \cdot 50 \cdot 2929=49,500,100

Extension Investigation

2. Why is this a Permutation problem, rather than a Combination problem? Explain.
It is a permutation problem because the order of the numbers and letters on the license plates makes a difference. The order of the numbers and letters is what makes a specific license plate identifiable. Imagine if you were asked the license plate number and you responded that you weren’t sure of the order but there was a 0, 5, B, K, A, K, and T. Would that be very helpful? No because there are a lot of different plates with those letters and numbers in any order. However, there is only one automobile with one specific order of the letters and numbers.
3. Why is this a Permutation with Repetitions problem? Explain.
It is a permutation with repetition problem because the letters and numbers can be repeated in the license plate, once a letter or number is used it is not off limits for further use in the license plate.
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.
See number 1 above.
b. Next, calculate how many two-digit year identifiers are possible.
See number 1 above.
c. Next, calculate how many two-letter area codes are possible.
See number 1 above.
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.
See number 1 above.
5. How would you approach this problem differently if letters and digits could not be repeated? Explain.
Then you would use the _nP_r formula as you cannot use the same letter or digit more than once. For example for the “area code” you would calculate _{26}P_2= 650 and then divide that by 2!=6, giving you a final answer of 325.
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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