Personalized License Plate: Part 2
MTH109 – Mathematical Explorations
Colorado State University – Global Campus
All vehicles in America must have a license plate in order to legally be driven on the roads. License plates vary from state to state in design and identification. Most states use a 6 character, unique identifier that consists of uppercase letters and numbers. In some states this is a random order and in others they use a combination of letters then numbers, or numbers then letters. Additionally, you can also apply for a customized plate which can have almost anything on it. If I wanted to personalize a plate for my new car that contained 6 characters of uppercase letters, A through Z, and numbers, 0 through 9, how many possible combinations could there be?
The goal of this report is to utilize probability theories, combinations, permutations and lots of multiplication to solve various questions that I could be presented with when creating a personalized license plate.
The Questions and Answers*
Assuming repetitions are allowed, how many custom tag numbers are possible?
Assuming repetitions are not allowed, how many custom tag numbers are possible?
Assuming repetitions are not allowed and there must be exactly three letters, how many custom tag numbers are possible?
Assuming repetitions are not allowed and there must be exactly three letters, located at the beginning of the license plate, how many custom tag numbers are possible?
Assuming repetitions are not allowed, there must be exactly three letters, located at the beginning of the license plate, and that the letters can only be vowels, how many passwords are possible?
*See Part 1 for full calculations
Answering each question proved to need an individual way to solve it using either permutations or combinations. Answers 1 through 3 required the use of combination formulas, while 4 and 5 needed permutation formulas to solve. Question 3 was the only one that required additional outside formulas to solve. For questions 1 and 2, the formulas for combinations with repetitions, C(n, r) = (r + n – 1)! / (r! × (n – 1)!), and combinations without repetitions, C(n, r) = n! / (r! × (n – r)!), were used. Question 3 used the same formula for question 1 but the value of n and r changed. For questions 4 and 5, the formulas for permutations were utilized in similar ways to find the different answers since both questions called for very specific information.
While solving each question I did come to see that there were more than one way to answering them. For example, question 1 basically wants to know how many combinations of 36 characters are there for a 6 character license plate, so another way to answer that would be 36×36×36×36×36×36 or 366. Because repetition does not matter this would mean that there would be a total of 2,176,782,336 combinations but that would include duplicates and/or arrangements that would not be technically be accepted if we were to really submit this to the DMV.
Using this same logic, it would also be possible to answer question 2 in a different manner. Question 2 asks how many combinations of customer plates there would be using 36 characters, but this time we can’t have repetition. Another way to calculate that would be 36×35×34×33×32×31. When we use this method of multiplication we are saying that the first character could be any of the 36 characters, multiplied by the remaining 35 characters, multiplied by the remaining 34 characters and so on. This would mean that there would be 1,402,410,240 combinations that did not include any repeats.
Repetitions v. No Repetitions
When using combinations and permutations it is important to understand the difference between the two. Combinations are distinct arrangements of a specified number of objects without regard to the order of selections from a specified set. Permutations, on the other hand, are an arrangement of objects where the order is important. Repetitions in either of these formulas are defined as double objects (Lippman, 2013).
In question 1 and 2, you can see that there is a large difference when really the only change in the questions was allowing or not allowing repetition. We know that question 1 will use a combination formula because the order does not matter and because repetition is allowed we know that we will use a formula that will take that into account. This is why we used the formula C(n, r) = (r + n – 1)! / (r! × (n – 1)!). When we allow for repetitions the number of possibilities increases because we can use the same characters multiple times. When repetitions are not allowed then we have to adjust the formula for that to show that we do not want to use the same characters more than once. This is why the answer for question 2 is smaller than the answer for question 1.
Solving Permutations and Combinations with Restrictions
Questions 3 and 4 required a bit more thought than what went into question 1 and 2 because 3 and 4 asked for specific information. In question 3, we wanted to make sure that there are exactly 3 letters in our custom tag, but the position of those letters is not defined like they are in question 4. Because all we have to do in question 3 is ensure that at least 3 of the characters are letters we have to change up how we solve this question. If we know that 3 characters will be letters and 3 characters will be numbers then we can determine that the number of times that letters can be used will be 263 or 17,576 times, and that the number of times a number can be used will be 103 or 1000 times. Since we only want 3 letters and 3 numbers and the order does not matter, we can utilize C(n, r) = n! / (r! × (n – r)!), where n = 6 and r = 3. This tells us that there are only 20 combinations that have 3 numbers and 3 letter used in any order.
When we restrict the locations of certain elements of a combination or permutation we significantly increase the number or possibilities. Because, in these problems, there are different elements being utilized (letters and numbers) we have to multiply those elements together in order to have all of the possibilities.
“A tree diagram is simply a way of representing a sequence of events. Tree diagrams are particularly useful in probability since they record all possible outcomes in a clear and uncomplicated manner” (Cork, 2011). We can use a tree diagram to answer question 5, since it is the least complex, in several different ways. One way that we could start is by drawing 5 “branches”, one for each vowel. Then, off of each vowel we could draw 4 more branches that represent the other 4 vowels not represented by the main branch. We would follow this by drawing 3 branches off of the 4 we just added that represent 3 of the vowels that were not represented by the first or second branch. This would give us the combination of possibilities that are available to use for the first 3 letters of the custom tag. This process would then continue for the numbers 0 through 9 so that we could see all of the possibilities available.
Probability and Reality
If we were really to use these methods to pick a custom license plate there would be a lot of issues that we would eventually run into. Although the guidelines in each state are different, the fact that there are guidelines are elements that have not been considered in answering these questions. In Colorado, we do use the 3 letter 3 number combination to generate license plates. This would have to be taken into consideration, and in order to properly determine what combinations we could submit we would have to know what combinations are not available. In addition to this, there are also restrictions as to what can be put on a license plate and the added factor that your custom tag can be more or less than 6 characters.
Advantages of Counting Techniques
There are many real life scenarios out there that counting techniques can be applied to. One of my favorite ones is deciding what to eat when out at a restaurant that offers many options for entrees, sides and desserts. Another scenario could be how a shop owner will determine which items to put up in a limited display window. Knowing how to utilize different counting techniques can help in both of those situations and many more. When you know what kind of counting method you can use, you can accurately know what your options are and then use them to the greatest benefit. It is a great way to be able to visualize the scenario before it goes into motion.
Permutations and combinations are a great way to see what possibilities are available to you when you are presented with different options. Permutations work best when there is an order you want to stick to. Combinations are great for creating unique passwords and seeing what in what ways you can organize things. With the questions and answers provided in this paper it is easy to see how counting techniques can be useful in many situations.
Cork, S. (2011, February). An Introduction to Tree Diagrams. Retrieved October 13, 2019, from https://nrich.maths.org/7288.
Lippman, D. (2013). Math in Society (2.4 ed.). Retrieved from http://www.opentextbookstore.com/mathinsociety/2.4/mathinsociety.pdf
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