COUNTING Rosen 4pt3 - Rosen 4.3 p324 1 List all the...

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Rosen 4.3 p324. 1. List all the permutations of {a, b, c}. The 3*2*1 ways to arrange these 3 elements: abc, acb, bac, bca, cab, cba 3. How many permutations {a, b, c, d, e, f, g} end with a? Count the permutations of {b, c, d, e, f, g} : 6*5*4*3*2*1 = 720 9. How many ways can 12 horses finish “in the money” if all orders of finish are possible? Calculate P(12, 3) = 12*11*10 = 1320 11. How many bit strings of length ten contain: a) exactly four 1s? We choose which four places contain those 1s : C(10, 4) = (10*9*8*7) / (4*3*2*1) = 210 b) at most four 1s? We add C(10, 4) + C(10, 3) + C(10, 2) + C(10, 1) + C(10, 0) = 386 c) at least four 1s? We subtract from the total number strings those that have only 0, 1, 2 or 3 1s : 2 10 – [ C(10, 0) + C(10, 1) + C(10, 2) + C(10, 3) ] = 1024 – 1 – 10 – 45 – 120 = 848 d) an equal number of 0s and 1s? Calculate C(10, 5) = 252 15. In how many ways can a set of five letters be selected from the English alphabet? C(26, 5) = 65780 17. How many subsets with more than two elements does a set with 100 elements have? We subtract from the total number of subsets those that have 0, 1, or 2 elements : 2 100
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