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Unformatted text preview: Solutions - Homework #3 1 Chapter 2 1.1 Problems 1. Problem 17 : If 8 castles (rooks) are randomly placed on a chessboard, compute the probability that none of the rooks can capture any of the others. That is, compute the probability that no row or file contains more than one rook. Since there are 64 squares on a chessboard with 8 rows and 8 columns, there are: 64 8 ! ways to place 8 rooks in 64 squares (assuming no two are in the same spot). With 8 rows & 8 columns, every row & every column must have exactly 1 rook. So choosing where to place a rook is equivalent to choosing a row/column pair. How many such pairs are possible? Choose row 1. There are 8 ways to choose a column as its mate. Choose row 2. There are 7 remaining ways to choose its column mate, etc. 8 7 2 1 = 8! ways to choose the pairs. P (No row/column with more than 1 rook) = 8! 64 8 ! = 9 . 109 10- 6 . 2. Problem 37 : An instructor gives her class a set of 10 problems with the information that the final exam will consist of a random selection of 5 of them. If a student has figured out how to do 7 of the problems, what is the probability that he or she will answer correctly: (a) All 5 problems? There are: 10 5 ! total ways to have a random selection of 5 out of 10 problems, 7 5 ! ways to have 5 of the 7 workable problems selected, and 3 ! = 1 way to have none of the unworkable problems selected....
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- Fall '07