License plates on cars are composed of up to 7 characters. The characters can either be a digit (0,1,...,9) or an uppercase lett
er (A,B,...,Z). For example, your license plate could be: AA04R32 or S9332. (Explain your answers for each part.) (a) (3 points) How many possible license plates are there that have at least one even number and at least one vowel? (b) (3 points) Since O’s and 0’s look similar, in some states, a valid license plate cannot have both an O and a 0. How many such 7-character license plates are there? (c) (3 points) How many possible 7-character license plates are there such that the first 3 characters are 3 distinct digits in increasing order and the remaining 4 characters are 4 distinct letters in alphabetical order? (d) (3 points) There is a particular 4-letter word that is not allowed on any license plate. (Let’s pretend that the word is MATH.) How many 7-character license plates avoid the word MATH? (e) (3 points) How many different 7-character license plates avoid having 2 digits in a row?
