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Unformatted text preview: MATH 55 Prof. Bernoff Fall 2007 Harvey Mudd College Homework 2 Solutions 7.6 In how many ways can a black rook and a white rook be placed on different squares of a chess board so that neither is attacking the other? (In other words, they cannot be in the same row or the same column of the chess board. A standard chess board is 8 8.) Solution: The only condition on the white rook is that it cannot be placed in the same row or column as the black one. There are 64 choices for where to place the black rook, and of the 63 remaining squares, 14 are in the same row or column, leaving 49. Thus there are 64 49 = 3 , 136 combinations. 7.10 A computer operating system allows files to be named using any combination of uppercase letters (A-Z) and digits (0- 9), but the number of characters in the file name is at most eight (and there has to be at least one character in the file name). For example, X23, W, 4AA, and ABCD1234 are valid file names, but W-23 and WONDERFUL are not valid (the first has an improper character and the second is too long). How many different file names are possible in this system? Solution: For each admissible length n (1 n 8) there are (26 + 10) n = 36 n combinations. In total there are 8 i =1 36 n = 36 9- 36 35 = 2 , 901 , 713 , 047 , 668 possible file names. 7.15 Four cards are drawn from a standard deck of 52 cards. In how many ways can this be done if the cards are all of different values (e.g., no two 5s or two jacks) and all of different suits? (For this problem, the order in which the cards are drawn matters, so drawing A -K -3 -6 is not the same as 6 -K -3 -A even though the same cards are selected.) Solution: Since order matters, there are 13 12 11 10 = 17160 possible ways to choose the values and 4 3 2 1 = 24 possible ways to choose the suits. In all there are 17160 24 = 411840 combinations....
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This homework help was uploaded on 04/22/2008 for the course MATH 55 taught by Professor Bernoff during the Fall '07 term at Claremont.
- Fall '07