License Plates

In the state of Comb in a country called Torics, the state government has recently discovered the existence of cars from Western civilization. They are excited to bring cars into the country; however, they must first come up with a system for selecting license plates before they can be driven on the roads. They want the system to be complicated enough to make it difficult to forge fake license plates but also simple enough to have enough license plates for the coming years.
Exercise 1: How many license plate are possible if each license plates if
  1. They must be length 6.
  2. Characters are from the usual English alphabet: A,B,C, \dots, Z.
Solution. There is no restrictions on repetition and the English alphabet has 26 letters. Therefore, we have 26^6 possible license plates.


Exercise 2: How many license plate are possible if each license plates if
  1. They must be length 7.
  2. Characters are from the usual English alphabet: A,B,C, \dots, Z or digits 0,1,2, \dots, 9.
Solution. There is no restrictions on repetition, the English alphabet has 26 letters. When we consider this with the digits, we have a total of 36 possible characters. Therefore, we have 36^7 possible license plates.


After reviewing some samples of license plate, they realize it is difficult to distinguish between 0 and O. They decide to remove the 0.

Exercise 3: How many license plate are possible if each license plates if
  1. They must be length 5.
  2. Characters are from the usual English alphabet: A,B,C, \dots, Z or digits 1,2, \dots, 9.
Solution. There is no restrictions on repetition, the English alphabet has 26 letters. When we consider this with the digits excluding zero, we have a total of 35 possible characters. Therefore, we have 35^5 possible license plates.


Exercise 4: How many license plate are possible if each license plates if
  1. They must be length k.
  2. Characters are from an alphabet of size n.
Solution. There is no restrictions on repetition, the alphabet has n letters, thus there are n^k possible license plates.


Exercise 5: Is there a difference between removing the 0 or the O from the possible characters?
Solution. No. It reduces the alphabet size by 1 in both cases.


After reviewing more samples of license plates, they found some license plates had too many repeating letters or digits, such as: “\text{OO}111” and “\text{AAAA}5”. They decide they should require some distinction between letters and digits. They start with some smaller sets:

Exercise 6: How many license plate are possible if each license plates if
  1. They must be length 3.
  2. Characters are from the digits 1, 2, 3, 4, 5.
  3. All characters must be distinct.
Solution. Each character must be distinct, therefore, for each option we choose, we have one less available for the next option. Therefore, we have 5\cdot 4 \cdot 3 possible license plates.


Exercise 7: How many license plate are possible if each license plates if
  1. They must be length 3.
  2. Characters are from the digits 1, 2, 3, 4, 5.
  3. All characters must not be repeated more than 2 times.
Solution. In the first choice we make, we have 5 options. Now, depending on our second choice, we will have a varying amount of possibilities for our third choice. If our second choice matches our first, then we have 4 remaining possibilities. And thus, in this case we have 5\cdot 1 \cdot 4 possibilities. In the second case, our second choice is different from our first (we have four possibilities here), and thus, we have no repeated any digit at this point. Therefore, we can choose any digit for the third choice, thus we have 5\cdot 4 \cdot 5. We add the counts for each case to get 5\cdot 1\cdot 4 + 5\cdot 4 \cdot 5 possible license plates.


Exercise 8: How many license plate are possible if each license plates if
  1. They must be length 4.
  2. Characters are from the digits 1, 2, 3, 4, 5.
  3. All characters must not be repeated more than 2 times.
Solution. We begin with 5 possible first choices.

Case 1: Choice 2 is the same as Choice 1. Then, we can select anything for our remaining 4 possibilities since selecting two more will not yield a repetition of more than 2 of the same character. Thus, in case 1, we have 5\cdot 1\cdot 4 \cdot 4 possibilities.

Case 2: Choice 2 is different from Choice 1. Then, we have to consider how make our third choice.

Case 2a: Choice 3 is the same as our Choice 1 or Choice 2. Then, one character is repeated twice, so we have four remaining options. Thus, we have 5\cdot 4 \cdot 2 \cdot 4 possibilities.

Case 2b: Choice 3 is different from Choice 1 and Choice 2. Then, no character is repeated, so we have five remaining options. Thus, we have 5\cdot 4\cdot 3 \cdot 5 possibilities.

Therefore, in Case 2, we have 5\cdot 4 \cdot 2 \cdot 4 + 5\cdot 4\cdot 3 \cdot 5 possibilities in case 2.

Adding the cases together, we have 5\cdot 1\cdot 4 \cdot 4 + 5\cdot 4 \cdot 2 \cdot 4 + 5\cdot 4\cdot 3 \cdot 5 possible license plates.