I want to choose a license plate which is 3 characters long, where the first character is a letter, the last character is a digit, and the middle is either a letter or a digit. I also want there to be two characters on my license plate which are the same. How many ways are there for me to choose a license plate with these restrictions?
I need to count how many different license plates I can make with 3 characters, where the first is a letter, the last is a digit, and the middle is either a letter or a digit. I also want there to be two characters on my license plate which are the same. I can use the multiplication principle to count the number of ways to choose each character. For the first character, I have 26 choices, since there are 26 letters in the alphabet. For the last character, I have 10 choices, since there are 10 digits from 0 to 9. For the middle character, I have 36 choices, since there are 26 letters and 10 digits. However, I also need to make sure that two characters are the same. This means that the middle character has to match either the first or the last character. So I have to consider two cases: either the middle character matches the first character, or it matches the last character. If the middle character matches the first character, then I have 26 choices for the first character, 1 choice for the middle character, and 10 choices for the last character. If the middle character matches the last character, then I have 26 choices for the first character, 10 choices for the middle character, and 1 choice for the last character. So the total number of ways to choose a license plate with these restrictions is the sum of these two cases: 26 x 1 x 10 + 26 x 10 x 1. This simplifies to 260 + 260 = 520. 