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55PS6 - SOLUTION SET 6DUE This solution set is currently...

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SOLUTION SET 6—DUE 3/12/2008 This solution set is currently only partially complete, to allow a post before the midterm. Please report any errors in this document to Ian Sammis ( [email protected] ). Problem 1 (#5.3.12) . How many bit strings of length 12 contain a) exactly 3 1s? b) at most three 1s? c) at least thee 1s? d) an equal number of 0s and 1s? Solution. In each case, once you know that you want m 1s, there are ( 12 m ) ways to arrange those 1s (you choose the positions of the 1s). Thus a) If there are exactly 3 1s, then there are ( 12 3 ) = 220 ways to place them into the bit string. Since this is a bit string, once the 1s have been placed the string is fully specified. b) Now, there are either 0, 1, 2, or 3 ones. Since those cases are disjoint, we can sum the counts, for ( 12 0 ) + ( 12 1 ) + ( 12 2 ) + ( 12 3 ) = 1 + 12 + 66 + 220 = 299 such strings. c) If there are at least three 1s, we’d have to sum over a lot of cases to use the technique of part (b). It’d be much easier to subtract off the cases that we aren’t interested in. There are 2 12 = 4096 bit strings of length 12, of which we don’t care about the ones with 0 ones (1 case), 1 one (12 cases), or 2 ones (66 cases). Thus there are 4096 - 1 - 12 - 66 = 4017 bit strings of length 12 with at least three 1s. d) If there are an equal number of 0s and 1s, there are 6 of each. There are ( 12 6 ) = 924 such bit strings. Problem 2 (#5.3.16) . How many subsets with an odd number of elements does a set with 10 elements have? Solution. We have to choose 1,3,5,7, or 9 elements of the 10 to be in our subset. There are ( 10 1 ) + ( 10 3 ) + ( 10 5 ) + ( 10 7 ) + ( 10 9 ) ways to do this. We can save ourselves a little bit of work by using the symmetry of the binomial coefficients to write this as 2 ( 10 1 ) + 2 ( 10 3 ) + ( 10 5 ) = 2(10) + 2(120) + 252 = 512 . Problem 3 (#5.3.26) . Thirteen people on a softball team show up for a game. a) How many ways are there to choose 10 players to take the field? b) How may ways are there to assign the 10 positions by selecting players from the 13 people who show up?

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