Exercise 8

In this exercise, we consider strings made from uppercase letters in the English alphabet and decimal digits. How many strings of length $10$ can be constructed in each of the following scenarios?

1. The first and last characters of the string are letters.

2. The first character is a vowel, the second character is a consonant, and the last character is a digit.

3. Vowels (not necessarily distinct) appear in the third, sixth, and eighth positions and no other positions.

4. Vowels (not necessarily distinct) appear in exactly two positions.

5. Precisely four characters in the string are digits and no digit appears more than one time.