1. How many 10-bit binary strings are there subject to each of the following res

Question

1. How many 10-bit binary strings are there subject to each of the following restrictions?

a. no restrictions

b. the string starts with 001

c. the string starts with 001 or 10

d. the string has exactly six 0's

e. the string has exactly six 0's and the first bit is 1

f. there is exactly one 1 in the first half and exactly three 1's in the second half

Explanation / Answer

Dear Student Thank you for using Chegg !! a) no restrictions Since no restriction then each bit has two options either 0 or 1 Hence total number of possible cases = 2^10 Solution = 1024 b) the string starts with 001 Since the string starts with 001 => three of the 10 bits are reserved => The rest 7 bits can be filled by ether 0 or 1 Total Number of permutation possible = 2^7 Solution = 128 c) the string starts with 001 or 10 c-1) For part 1) refer to solution b above = 128 c-2) Since the string starts with 10 => two of the 10 bits are reserved The rest 8 bits can be filled by ether 0 or 1 Total Number of permutation possible = 2^8 Solution = 256 Total no of solution = 2^7 + 2^8 = 384 d) the string has exactly six 0's Since the string has exactly 6 '0', hence the rest 4 must be 1 So now the string has 6 '0' and 4 '1s and these 0's and 1's are arranged to find out the number of permutations Total number of permutation = 10! /(6!4!) (6 0's & 4 1's are of a kind) = 210 Kindly note that at chegg we have a policy of answering 1 question at a time. Thank you !!

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