TDOT MATHEMATICAL ASSOCIATION OF AMERICA WeBWorK Math151-McLaughlin DM5.2 8 M5 2
ID: 3147078 • Letter: T
Question
TDOT MATHEMATICAL ASSOCIATION OF AMERICA WeBWorK Math151-McLaughlin DM5.2 8 M5 2. Poblem pt) How many B-bit strings (that is, bit strings of length 8) are there which: 1. Start with the sub-string 101? 2. Have weight 5 (i.e, contain exactly five I's) and start with the sub-string 101? 3. Either start with 101 or end with 11 (or both)? 4. Have weight 5 and either start with 101 or end with 11? ote: You can earn partial credit on this problem Preview Answers Submit Answers ou have attempted this problem 0 times. our overall recorded score is 0%. ou have unlimited attempts remaining l instructor ESTExplanation / Answer
1. Since the first three digits are 101, we need to check for the next five digits.
Each digit has two options - 0 and 1.
=> Number of numbers = 25 = 32.
2. Since 101 has two 1s, the remaining five digits have 3 ones and 2 zeroes.
The 3 ones can occupy 5C3 = 10 places. Remaining two places will be occupied by zeros.
=> Number of numbers = 10.
3. If the number ends with 11, remaining six digits have 26 = 64 choices.
If the number begins with 101 and ends with 11, the 3 digits in between have 23 = 8 choices.
=> Number of numbers which begin with 101 or end with 11 = 32 + 64 - 8 = 88.
4. If the number ends with 11 and has weight 5, there are 3 ones and 3 zeros among the remaining 6 digits and this can happen in 6C3 = 20 ways.
If the number begins tiwh 101, ends with 11 and has weight 5, there is 1 one among the remaining 3 digits. This can happen in 3 ways.
=> Number of numbers which begin with 101 or end with 11 and have weight 5 = 10 + 20 - 3 = 27.
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