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4. Consider bit strings with lenght 1 and weight k (so strings of 1 0's and 1's, including k 1's). We know how to count the number of these for a fixed 1 and k. Now, we will count the number of strings for which the sum of the length and the weight is fixed. For example, let's count all the bit strings for which 1+ k = 11. (a) Find examples of these strings of different lengths. What is the longest string possible? What is the shortest? (b) How many strings are there of each of these lengths. Use this to count the total number of strings (with sum 11). (c) The other approach: Let n = 1+ k vary. How many strings have sum n = 1? How many have n = 2? And so on. Find and explain a recurrence relation for the sequence (an) which gives the sumnumber of strings with sum n. (d) Describe what you have found above in terms of Pascal's Triangle. What patter have you discovered?

4. Consider bit strings with lenght 1 and weight k (so strings of 1 0's and 1's, including k 1's). We know how to count the number of these for a fixed 1 and k. Now, we will count the number of strings for which the sum of the length and the weight is fixed. For example, let's count all the bit strings for which 1+ k = 11. (a) Find examples of these strings of different lengths. What is the longest string possible? What is the shortest? (b) How many strings are there of each of these lengths. Use this to count the total number of strings (with sum 11). (c) The other approach: Let n = 1+ k vary. How many strings have sum n = 1? How many have n = 2? And so on. Find and explain a recurrence relation for the sequence (an) which gives the sumnumber of strings with sum n. (d) Describe what you have found above in terms of Pascal's Triangle. What patter have you discovered?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. Consider bit strings with lenght 1 and weight k (so strings of 1 0's and 1's, including k 1's). We know how to count the number of these for a fixed 1 and k. Now, we will count the number of strings for which the sum of the length and the weight is fixed. For example, let's count all the bit strings for which 1+ k = 11. (a) Find examples of these strings of different lengths. What is the longest string possible? What is the shortest? (b) How many strings are there of each of these lengths. Use this to count the total number of strings (with sum 11). (c) The other approach: Let n = 1+ k vary. How many strings have sum n = 1? How many have n = 2? And so on. Find and explain a recurrence relation for the sequence (an) which gives the sum number of strings with sum n. (d) Describe what you have found above in terms of Pascal's Triangle. What patter have you discovered?
Transcribed Image Text:4. Consider bit strings with lenght 1 and weight k (so strings of 1 0's and 1's, including k 1's). We know how to count the number of these for a fixed 1 and k. Now, we will count the number of strings for which the sum of the length and the weight is fixed. For example, let's count all the bit strings for which 1+ k = 11. (a) Find examples of these strings of different lengths. What is the longest string possible? What is the shortest? (b) How many strings are there of each of these lengths. Use this to count the total number of strings (with sum 11). (c) The other approach: Let n = 1+ k vary. How many strings have sum n = 1? How many have n = 2? And so on. Find and explain a recurrence relation for the sequence (an) which gives the sum number of strings with sum n. (d) Describe what you have found above in terms of Pascal's Triangle. What patter have you discovered?
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