Exercise 3

Matt is designing a website authentication system. He knows passwords are most secure if they contain letters, numbers, and symbols. However, he doesn't quite understand that this additional security is defeated if he specifies in which positions each character type appears. He decides that valid passwords for his system will begin with three letters (uppercase and lowercase both allowed), followed by two digits, followed by one of $10$ symbols, followed by two uppercase letters, followed by a digit, followed by one of $10$ symbols. How many different passwords are there for his website system? How does this compare to the total number of strings of length $10$ made from the alphabet of all uppercase and lowercase English letters, decimal digits, and $10$ symbols?