Homework Answers
a.
Answer: 210
Each bit can be 0/1 -- 2 possibilities
For 10 bits it is 210
b.
Answer: 26
4 bits are to start with 1101
Remaining are 6 bits which have 26 possibilities
c.
Answer: 210
Exactly 6 zeros
Remaining 4 are ones
To place 6 zeros in 10 different places, possibilities are 10C6
C(10, 6) = 210
d.
Answer: 252
Equal number of 0s and 1s -- 5 each is the only
possibility
To place 5 zeros/ones in the 5 palces = 10C5
C(10, 5) = 252
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Let A be the set of all bit strings of length 10. 1. How many bit...
Problem 3 (Counting binary strings) 20 marks/ Consider all bit strings of length 15. 1. How many ...
Problem 3 (Counting binary strings) 20 marks/ Consider all bit strings of length 15. 1. How many begin with 00? 2. How many begin with 00 and end with 11? 3. How many begin with 00 or end with 10? 4. How many have exactly ten 1's? 5. How many have exactly ten 1's such as none of these 1's are adjacent to each other? Provide detailed justifications for your answers.
Problem 3 (Counting binary strings) 20 marks/ Consider all...
Consider all bit strings of length 12. How many have exactly four 1's?
Consider all bit strings of length 12. How many have exactly four 1's?
Consider all bit strings of length 12 How many have 8 0s and 4 1s that have exactly 3 consecutive...
consider all bit strings of length 12 How many have 8 0s and 4 1s that have exactly 3 consecutive 1s (allow 4 consecutive 1s)? A 90 В 72 С 45 D 36 E 9
consider all bit strings of length 12 How many have 8 0s and 4 1s that have exactly 3 consecutive 1s (allow 4 consecutive 1s)? A 90 В 72 С 45 D 36 E 9
Multiple Choice How many strings of length 12 over the alphabet {a,b,c} have exactly three a's...
Multiple Choice How many strings of length 12 over the alphabet {a,b,c} have exactly three a's or have exactly three b's or have exactly three c's? (1?).22-3-(3) °(12):22-3-(?) (3) ° (13)-3-(1) QUESTION 20 Multiple Choice How many binary strings of length 12 have exactly six 1's or begin with a 0? ° (62) +211 -(0 ° (12) +201 - 1 ° (6) +211 -(5) ° (12) + 211
How many bit strings of length n are palindromes
How many bit strings of length n are palindromes? Hint: Consider two cases n is even and n is odd. Note a palindrome is a “string” of letters or numbers which read the same “frontwards” and backwards”. Examples: 1101011, 10111101 are palindromes.
6. (10 points) Let Sn be the number of n-bit strings of O's and 1's that...
6. (10 points) Let Sn be the number of n-bit strings of O's and 1's that avoid the pattern "11". Find the recurrence relation for sm and find the solution for this recurrence relation.
1. Use a Regular Expression to define the set of all bit strings of one or...
1. Use a Regular Expression to define the set of all bit strings
of one or more 0's followed by only a 1.
2. Use a Regular Expression to define the set of all bit string
of two or more symbols followed by three or more 0's.
3. Are these two grammars the same?
a. S-> aSb|ab|λ
b. S-> aAb|ab A->aAb|λ
4. Use the process of elimination to find the language of the
following FA: (see picture for diagram)
5....
Exercise 8.12.20: Counting binary strings. (a) How many binary strings of length 12 do not have...
Exercise 8.12.20: Counting binary strings. (a) How many binary strings of length 12 do not have exactly four 1's? (b) How many binary strings of length 12 start with 101 or 1110? (e) How many binary strings of length 12 start with 00 or end with 00 or both?
1.1 Let S = {01, 10, 11}. Note that S is a set of 2-bit strings...
1.1 Let S = {01, 10, 11}. Note that S is a set of 2-bit strings with string 00 missing. Consider the following three One-Time Pad (OTP) variants. For each of these OTP variants state whether the resulting cipher is perfectly secure or not, and prove your answer. In other words, if your answer is “yes”, prove that the cipher passes Shannon’s perfect secrecy criterion, and if your answer is “no” then show that the cipher fails this criterion. In...
Problem 3 a) How many strings are there of length 10 over the alphabet (a, b)...
Problem 3 a) How many strings are there of length 10 over the alphabet (a, b) with exactly five a's? b) How many strings are there of length 10 over the alphabet (a, b, c) with exactly five a's?
Problem 3 (Counting binary strings) 20 marks/ Consider all bit strings of length 15. 1. How many begin with 00? 2. How many begin with 00 and end with 11? 3. How many begin with 00 or end with 10? 4. How many have exactly ten 1's? 5. How many have exactly ten 1's such as none of these 1's are adjacent to each other? Provide detailed justifications for your answers.
Problem 3 (Counting binary strings) 20 marks/ Consider all...
consider all bit strings of length 12 How many have 8 0s and 4 1s that have exactly 3 consecutive 1s (allow 4 consecutive 1s)? A 90 В 72 С 45 D 36 E 9
consider all bit strings of length 12 How many have 8 0s and 4 1s that have exactly 3 consecutive 1s (allow 4 consecutive 1s)? A 90 В 72 С 45 D 36 E 9
Multiple Choice How many strings of length 12 over the alphabet {a,b,c} have exactly three a's or have exactly three b's or have exactly three c's? (1?).22-3-(3) °(12):22-3-(?) (3) ° (13)-3-(1) QUESTION 20 Multiple Choice How many binary strings of length 12 have exactly six 1's or begin with a 0? ° (62) +211 -(0 ° (12) +201 - 1 ° (6) +211 -(5) ° (12) + 211
6. (10 points) Let Sn be the number of n-bit strings of O's and 1's that avoid the pattern "11". Find the recurrence relation for sm and find the solution for this recurrence relation.
1. Use a Regular Expression to define the set of all bit strings
of one or more 0's followed by only a 1.
2. Use a Regular Expression to define the set of all bit string
of two or more symbols followed by three or more 0's.
3. Are these two grammars the same?
a. S-> aSb|ab|λ
b. S-> aAb|ab A->aAb|λ
4. Use the process of elimination to find the language of the
following FA: (see picture for diagram)
5....
Problem 3 a) How many strings are there of length 10 over the alphabet (a, b) with exactly five a's? b) How many strings are there of length 10 over the alphabet (a, b, c) with exactly five a's?
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