1. ## Bit string question

How many 8 bit strings either start with a 1 or end with a one?

How many 8 bit strings either start with 100 or have the fourth bit a 1?
I know that the answer is 2^5 + 2^7 - 2^4 The first term is the number of strings with 100 the second term is the number of strings with 1 as 4th bit but why is 2^4 the number in which both occur??? Is it simply that there are 2^4 ways to choose four positions???? Thank you all for looking and trying to help my old brain think in new ways.

2. $\displaystyle \begin{array}{rcl} {\left| {S \cup E} \right|} & = & {\left| S \right| + \left| E \right| - \left| {S \cap E} \right|} \\ {} & = & {2^7 + 2^7 - 2^6 } \\ \end{array}$

3. ## Bit string question

Plato, thank you for your explanation. I am a bit confused though. If you have to have 100 then are you not just picking the other 5 bits so the first term would be 2^5??

4. Hello, Frostking!

. . (I bet my brain is older than yours.)

How many 8-bit strings either start with a 1 or end with a 1?

If it starts with 1, it is of the form: .$\displaystyle 1\,\_\,\_\,\_\,\_\,\_\,\_\,\_$
. . The blanks can be filled in $\displaystyle 2^7$ ways.

If it ends with 1, it is of the form: .$\displaystyle \_\,\_\,\_\,\_\,\_\,\_\,\_\,1$
. . The blanks can be filled in $\displaystyle 2^7$ ways.

But there are numbers that start and end with 1,
. . and we have counted them twice.
These numbers have the form: .$\displaystyle 1\,\_\,\_\,\_\,\_\,\_\,\_\,1$
. . The blanks can be filled in $\displaystyle 2^6$ ways.

Therefore, the number is: .$\displaystyle 2^7 + 2^7 - 2^6 \;=\;\boxed{192}$

How many 8-bit strings either start with 100 or have the fourth bit a 1?

If it starts with 100, it is of the form: .$\displaystyle 100\,\_\,\_\,\_\,\_\,\_$
. . The blanks can be filled in $\displaystyle 2^5$ ways.

If the fourth digit is 1, it is of the form: .$\displaystyle \_\,\_\,\_\,1\,\_\,\_\,\_\,\_$
. . The blanks can be filled in $\displaystyle 2^7$ ways.

But there are numbers that begin with 100 and whose 4th digit is 1,
. . and we have counted them twice.
These numbers are of the form: .$\displaystyle 1001\,\_\,\_\,\_\, \_$
. . The blanks can be filled in $\displaystyle 2^4$ ways.

Therefore: .$\displaystyle 2^5 + 2^7 - 2^4 \;=\;\boxed{144}$

5. ## Bit string question

Thank you very much!!! With that explanation I think I can do just about any bit problem that is given to me. I appreciate your being willing to explain it at such a basic level and stating it so clearly. You do have me in age but not by much!! Have a great day!