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Math245Quiz6Sum00 - c Find the number of decimal strings of...

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Math 245 KEEP THIS SHEET!!! QUIZ 6 SECTIONS 5.1, 5.2, 5.5: RECURRENCE RELATIONS AND INCLUSION-EXCLUSION Show all work where appropriate! Write your name on all of your worksheets. For each problem, clearly indicate the problem number and/or letter, relevant work, and your answer. Note: Unless otherwise specified, "or" means "inclusive or". ------------------------------------------------------------------------------------ 1) Find a recurrence relation for the number of bit strings of length n that contain four consecutive "1"s. (10 points) ------------------------------------------------------------------------------------ 2) A decimal string of length n consists of n digits. (A "digit" may be 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9). (25 points total) a) Find a recurrence relation for the number of decimal strings of length n that do not contain three consecutive "0"s. (10 points) b) Find the number of decimal strings of length 3 that do not contain three consecutive "0"s. (5 points)
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Unformatted text preview: c) Find the number of decimal strings of length 4 that do not contain three consecutive "0"s. (10 points)------------------------------------------------------------------------------------3) Solve the recurrence relation a n = 7 a n − 1 − 6 a n − 2 n ≥ 2 ( ) together with the initial conditions a = 30 a 1 = 20 ⎧ ⎨ ⎪ ⎩ ⎪ . (20 points)------------------------------------------------------------------------------------4) Use an iterative approach to find the solution (as an explicit, closed-form formula) to the recurrence relation a n = a n − 1 + 3 n + 2 n ≥ 1 ( ) together with the initial condition a = 5 . (20 points)------------------------------------------------------------------------------------5) Use the inclusion-exclusion principle to find the number of decimal strings of length 5 that: • begin with three even digits, or • begin with a "0", or • end with three even digits. Show all work! (25 points)...
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