17. How many 10-digit binary strings are there that have exactly four l's or exactly five 1's? How many do not have exactly four I's or exactly five I's?

In how many ways can we arrange all the letters in the word INFORMATION so that no pair of consecutive letters occurs more than once? (Note: IINNOOFRMTA is allowed but not INFORINMOTA, as IN occurs twice)

1. 2. Consider the set of words (character strings) consisting "a" and "b", e.g. "aaba", "bbaba". (a) How many words have length 10 in this set? (b) A word is called special if it is lexicographically strictly smaller than any of its proper suffixes. For example, "ababb" is special. How many words of length 10 are special? Suppose that two identical decks of 52 distinct cards (total 104 cards) are mixed together and are randomly divided into 13 hands of 8 cards each. What is the probability that no hand has more than one ace?

5. How many 16-digit binary strings contain exactly seven l's? (Examples of such strings include 0111000011110000 and 0011001100110010, etc.)

7. What is the minimum number of students, each of whom comes from one of the 50 states, who must be enrolled in a university to guarantee that there are at least 100 who come from the same state? 8. How many permutations of the letters ABCDEFG contain (a) the string CFGA? (b) the strings BA and GF?

(a) How many bit strings of length nine either begin with two 0s or end with three 1s?(b) How many different four-letter initials with none of the letters repeated can people have?(c) How many permutations of the letters WXYZABCD contain the strings ZAB and WD?(d) How many different strings can be made from the letters in HAMHANDEDNESSES, usingall the letters?

8. If 100 positive integers are randomly selected, at least how many of them are guaranteed to have the same remainder when divided by 7? 9. At least how many positive integers must be selected in order to guarantee (a) At least two of them have the same remainder when divided by 7? (b) At least five of them have the same remainder when divided by 7?

16) How many binary strings of length 12 have exactly six 1's or begin with a 0?

13. How many length-6 lists can be made from the symbols {A, B, C, D, E, F, G), if repetition is allowed and the list is in alphabetical order? (Examples: BBCEGG, but not BBBAGG.)

15. How many 10-digit binary strings begin in 1 or end in 1?