Math245Sols6Sum00

Math245Sols6Sum00 - MATH 245: QUIZ 6 SOLUTIONS 1) Let a n =...

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Unformatted text preview: MATH 245: QUIZ 6 SOLUTIONS 1) Let a n = the number of bit strings of length n with at least one occurrence of "1111". Consider the following disjoint cases: Case 1: length n 1 a n-1 possibilities Case 2: 10 length n 2 a n-2 possibilities Case 3: 110 length n 3 a n-3 possibilities Case 4: 1110 length n 4 a n-4 possibilities Case 5: 1111 length n 4 2 n-4 possibilities, since the rest of the string can be any sequence of n-4 bits. Answer: a n = a n 1 + a n 2 + a n 3 + a n 4 + 2 n 4 2) a) Let a n = the number of decimal strings of length n with no occurrences of "000". Case 1: 9 possibilities can be 1-9 length n 1 9 a n-1 possibilities Case 2: 1 "0" 9 can be 1-9 length n 2 9 a n-2 possibilities Case 3: 1 "0" 1 "0" 9 can be 1-9 length n 3 9 a n-3 possibilities...
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Math245Sols6Sum00 - MATH 245: QUIZ 6 SOLUTIONS 1) Let a n =...

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