Only answer the question 8 and 9.

A dec-string is a sequence of characters from the 10-character alphabet {0,1,2,3,4,5,6,7,8,9}. For example, these are dec-strings: 36562342320 49548362729 Let n > O be an integer. 1. What is the number of dec-strings of length n? 2. What is the number of dec-strings di,...,n of length n such that did2 +00. In other words, what is the number of dec-strings of length n that don't begin with 00? 3. What is the number of dec-strings di, ..., dn of length n such that did2 700 and d2d3 # 11? 4. What is the number of dec-strings di,...,n of length n such that did2 + 00 and d2d3 + 01? 5. What is the number of dec-strings d1,..., dn of lenght n such that did2 = 00 or did2d3 = 111? 6. What is the number of dec-strings di,..., dn of length n > 4 such that did2 + 00 or d3d4 # 11. 7. A dec-string d,..., dn is bad if d = d;+1 or di+di+1 = 9 for at least one i {1,...,n-1} and it is good otherwise. What is the number of good dec-strings of length n? 8. A dec-string d1,..., dn is 2-bad if, d; = d; or di +d; = 9 for some i O be an integer. 1. What is the number of dec-strings of length n? 2. What is the number of dec-strings di,...,n of length n such that did2 +00. In other words, what is the number of dec-strings of length n that don't begin with 00? 3. What is the number of dec-strings di, ..., dn of length n such that did2 700 and d2d3 # 11? 4. What is the number of dec-strings di,...,n of length n such that did2 + 00 and d2d3 + 01? 5. What is the number of dec-strings d1,..., dn of lenght n such that did2 = 00 or did2d3 = 111? 6. What is the number of dec-strings di,..., dn of length n > 4 such that did2 + 00 or d3d4 # 11. 7. A dec-string d,..., dn is bad if d = d;+1 or di+di+1 = 9 for at least one i {1,...,n-1} and it is good otherwise. What is the number of good dec-strings of length n? 8. A dec-string d1,..., dn is 2-bad if, d; = d; or di +d; = 9 for some i