Questions #6 and #12

6. Consider lists made from the symbols A, B, C, D, E, with repetition allowed. (a) How many such length-5 lists have at least one letter repeated? (b) How many such length-6 lists have at least one letter repeated? 7. A password on a certain site must be five characters long, made from letters of the alphabet, and have at least one upper case letter. How many different passwords are there? What if there must be a mix of upper and lower case? 8. This problem concerns lists made from the letters A, B, C, D, E, F, G, H, I, J. (a) How many length-5 lists can be made from these letters if repetition is not allowed and the list must begin with a vowel? (b) How many length-5 lists can be made from these letters if repetition is not allowed and the list must begin and end with a vowel? (c) How many length-5 lists can be made from these letters if repetition is not allowed and the list must contain exactly one A? 9. Consider lists of length 6 made from the letters A, B, C, D, E, F, G, H. How many such lists are possible if repetition is not allowed and the list contains two consecutive vowels? 10. Consider the lists of length six made with the symbols P, R, O, F, S, where repetition is allowed. (For example, the following is such a list: (P,R,O,O,F,S).) How many such lists can be made if the list must end in an S and the symbol O is used more than once? 11. How many integers between 1 and 1000 are divisible by 5? How many are not divisible by 5? 12. Six math books, four physics books and three chemistry books are arranged on a shelf. How many arrangements are possible if all books of the same subject are grouped together?